3 Matrix Algebra. 3.1 Operations on matrices

Size: px
Start display at page:

Download "3 Matrix Algebra. 3.1 Operations on matrices"

Transcription

1 3 Matrix Algebra A matrix is a rectangular array of numbers; it is of size m n if it has m rows and n columns. A 1 n matrix is a row vector; an m 1 matrix is a column vector. For example: The (i, j)-entry of a matrix is the number in the ith row and the jth column. If we want to discuss a general m n matrix A we will denote its (i, j)-entry by a i j, so that a 1 1 a 1 2 a 1 3 a 1 n a 2 1 a 2 2 a 2 3 a 2 n A.... a m 1 a m 2 a m 3 a m n A square matrix is one of size n n, i.e. it has the same number of rows as columns. Two matrices A and B are equal if they are of the same size, and their corresponding entries are equal. 3.1 Operations on matrices Matrices A and B can be added if they are of the same size; the sum A + B is formed by adding the corresponding entries, so that the (i, j)-entry of A + B is a i j + bi j. For example [ 2 3 ] The zero matrix (of any given size m n) is denoted 0, and has only 0 as its entries. It satisfies A A A. For example If A is any matrix and k any number, then the scalar multiple ka is defined by multiplying each entry of A by k. For example By definition A means ( 1)A, so that A B means A + ( 1)B, the matrix obtained by subtracting the entries of B from those of A. Properties of these two operations include: A + B B + A; (A + B) + C A + (B + C); k(a + B) ka + kb; (k + l)a ka + la; (kp)a k(pa); A A 0; 0.A 0, provided that A, B and C are all of the same size. Note that 0 is used both as a scalar and as matrix in the last identity. If A is an m n matrix then the transpose of A, denoted A T, is the n m matrix whose rows are the columns of A written in the same order. For example T and T

2 MATRIX ALGEBRA Note that the (i, j)-entry of A T is the (j, i)-entry of A. Properties of the transpose operation include: (A T ) T A; (ka) T ka T ; (A + B) T A T + B T. The matrix A is symmetric if A A T. In this case A must have as many rows at it has columns, so is square. Exercise 3.1. Solve the following equations in each case to find the matrix A: (i) A T A T (ii) 3A + 2A T

3 Operations on matrices Exercise 3.2. Find the general form of a 2 2 symmetric matrix. The final operation we shall discuss is matrix multiplication. If A is a k m matrix and B is an m n matrix then their product is the k n matrix whose (i, j)-entry is computed by multiplying each entry of row i of A by the corresponding entry of column j of B and summing the results. That is, the (i, j)-entry of AB is the dot product of row i of A with column j of B. For example if A and B then 2 7 AB 1 8 ] 3 2 [ ( 2) ( 2) ( 2) ( 2) ( 2) Unlike our earlier operations involving two or more matrices, we no longer require A and B to be the same size. Above, A is a 3 2 matrix and B is a 2 2 matrix, so we can form the product AB, which is a 3 2 matrix. However, B does not have the same number of columns as A has rows (equivalently, the rows of B and the columns of A are of different lengths), so BA is not defined in this case. Matrices A and B are said to commute if AB BA. In particular if A is an m n matrix and B a p q matrix, then for the first product to be defined we need n p and for the second product we need q m. The product AB is then an m q matrix and BA is a p n matrix, and if AB BA they must be the same size, hence m p n q, that is, A and B are both square and of the same size. However given any two square n n matrices A and B, it need not be true that AB BA. For example: A, B AB but BA The identity matrix of size n n is the matrix with 1 appearing down the (main) diagonal, and 0 elsewhere. We write this as I, or I n if we want to specify the size. So 1 0 I 2, I ,

4 MATRIX ALGEBRA Note that In general, if A is an m n matrix then I m A AI n A. Other properties of matrix multiplication include A(BC) (AB)C; A(B + C) AB + AC; (A + B)C AC + BC; k(ab) (ka)b; whenever these products are defined. (AB) T B T A T Exercise 3.3. Solve the following equations to find the matrix A: (i) 3A (ii) 2A A T

5 Linear systems: the matrix viewpoint 3.2 Linear systems: the matrix viewpoint Consider the following system of linear equations: 2x 1 3x 2 + x 3 2 x 1 + x 2 + 3x 3 0 These two equations can be combined into a single equation about 1 2 column vectors: [ 2x1 3x 2 + x x 1 + x 2 + 3x 3 0] x 1 [ x ] That is, our system of equations is the same as the single equation AX B, where A, X x 1 [ x and B 0] This can be done for any system of m equations in n unknowns, where A is the m n matrix of coefficients, X the n 1 matrix of unknowns, and B the m 1 matrix of constants. Theorem 3.4. Suppose a linear system of equations is written as AX B, as above, and suppose that X 1 is a solution of this system. Any other solution X 2 of AX B is of the form X 2 X 1 + X 0 where X 0 solves the associated homogeneous system AX 0; conversely any vector of the form of X 2 is a solution of AX B. Proof. Suppose that X 1 and X 2 both solve the given (nonhomogeneous) system. That is, AX 1 B and AX 2 B. Then x 3 A(X 2 X 1 ) AX 2 AX 1 B B 0, and so X 0 : X 2 X 1 is a solution of the associated homogeneous such that X 2 X 1 + X 0. Conversely, suppose that X 0 solves AX 0, and that X 1 solves AX B. Let X 2 X 1 + X 0, then AX 2 A(X 1 + X 0 ) AX 1 + AX 0 B + 0 B, so that X 2 is a solution of the nonhomogeneous system. What this result tells us is that to find all solutions to a given linear system AX B it is enough to find just one solution and then find all solutions to the equation AX 0. We shall see this principle again in Section 6. Recall that for the homogeneous system AX 0 there is always at least one solution, namely X 0. If there is more than one, that is if the rank r of A is smaller than the number of columns n, then there are infinitely solutions given by varying the n r parameters. In this case we can find n r nonzero vectors X 1, X 2,...,X n r such that any solution of AX 0 can be written as t 1 X 1 + t 2 X t n r X n r for numbers t 1, t 2,...,t n r. Moreover these vectors have the property that t 1 X 1 + t 2 X t n r X n r 0 only if t 1 t 2 t n r 0. This situation is described by saying that the set of vectors X 1, X 2,..., X n r is linearly independent. An example of linearly independent vectors are the basis vectors i,j,k in R 3, since t 1 i + t 2 j + t 3 k (t 1, 0, 0) + (0, t 2, 0) + (0, 0, t 3 ) (t 1, t 2, t 3 ) and this equals (0, 0, 0) only if t 1 t 2 t 3 0. x 3 30

6 MATRIX ALGEBRA Remarks. (i) An alternative but equivalent definition for linear independence is that the set of vectors X 1, X 2,..., X n r is linearly independent if whenever X is any vector that can be written as a linear combination of X 1, X 2,..., X n r, then it can be done so in only one way. Again, this is clear for the standard basis vectors i,j,k, where the position vector for any point in R 3 can be written uniquely as a linear combination of these three. (ii) It follows that any two vectors X 1 and X 2 are linearly independent if and only if they are not parallel, i.e. they are not multiples of one another. Exercise 3.5. Show that x 1 3, x 2 4, x 3 2, x 4 0 and x 5 1 is a solution of the system 2x 1 + x 2 2x 3 + 3x 4 x x 1 2x 2 + 7x 4 + 5x 5 6 4x 1 5x 2 + 2x x x 5 1 Find the general solution by solving the associated homogeneous system. 31

7 Exercises 3.3 Exercises Let A, B 0 1 following (where possible): 3 1 2, C and D Compute the 1 4 (i) 3A 2B (ii) 5C (iii) 4A T 3C (iv) B + D (v) (A + C) T (vi) A D 2. Find A if (i) 5A 3A ( ) T (iii) 3A T [ [ 2 3 (ii) 3A + 5A 2 1] 0] (iv) ( 2A T 5 ) T 1 0 4A Compute the following matrix products (if possible): (i) (ii) (iv) a b 0 a b (v) 0 0 c 0 0 c (iii) (vi) In both cases express every solution of the given system as the sum of a specific solution plus a solution of the associated homogeneous system: (i) x y 4z 4 x + 2y + 5z 2 x + y + 2z 0 5. Let A, B and C be matrices. (ii) 2x 1 + x 2 x 3 x 4 1 3x 1 + x 2 + x 3 2x 4 2 x 1 x 2 + 2x 3 + x 4 2 2x 1 x 2 + 2x 4 3 (a) If A 2 can be formed, what can be said about the size of A? (b) If AB and BA can both be formed, describe the sizes of A and B. (c) If ABC can be formed, A is 3 3 and C is 5 5, what size is B? 3.4 Matrix inverses If a and b are numbers with a 0 then it is easy to solve the equation ax b just divide both sides by a to get x b a. Note also that this is the only solution. In the case when a 0, the equation ax b can only be solved if b 0 as well, in which case any value of x R is a solution. If a 0 and b 0 then there are no solutions. Now any system of m linear equations in n unknowns can be written as AX B, where A is m n, X is n 1 and B is m 1. By analogy with the above we would like to solve this system by dividing by A; however matrix division does not make sense. In the one variable case recall that when a 0 we define a 1 1 a, so that the solution of ax b is x a 1 b. Definition 3.6. An n n matrix A is invertible if there is another n n matrix B such that B is called the inverse of A. AB BA I. 32

8 MATRIX ALGEBRA Remark. In the definition we assumed that A is a square matrix, and consequently B is a square matrix of the same size. To see why we do this, suppose it were possible to find an m n matrix A and a n m matrix B such that m < n and AB I m, BA I n. Since A has more columns than rows, its rank is no greater than m, and hence less than n, thus there is a nontrivial solution X to the equation AX 0. But then X I n X (BA)X B(AX) B0 0, contradicting the fact that X 0. So we must have m n. But then, by symmetry, it follows that m n as well, so that m n, and hence it only make sense to talk about inverses of square matrices. Consider the following matrices: We have AB [ A [ ][ ] ], B [ ] 0 1 BA. 1 3 That[ is, B is ] the inverse of A, and A is the inverse of B. However consider the matrix 0 0 C. This has no inverse since 1 2 a b 0 0 a b 0 0 C I, c d 1 2 c d a 2b c 2d and there is no way to make the top left entry of the product equal to 1. So, unlike numbers, it is possible to have noninvertible matrices that are nonzero. However, given a nonzero number a 0, its multiplicative inverse is a 1 1 a it only has the one inverse. The same is true for matrices: Proposition 3.7. Any n n matrix has at most one inverse. Proof. Suppose that A, B and C are n n matrices such that Then AB I and CA I. ( ) C CI C(AB) (CA)B IB B. Since the equations in ( ) must hold if B and C are inverses of A, it follows that B and C must be the same, and so there can only be at most one inverse to A. Since if A is invertible it has only one inverse we can denote it by A 1, and talk about the inverse of A. Theorem 3.8. If A is an invertible n n matrix then for any n 1 vector B there is a unique solution to AX B, namely A 1 B. Proof. Consider the column vector X 1 A 1 B. This satisfies AX 1 A(A 1 B) (AA 1 )B IB B, and so is a solution. On the other hand, if X 2 is any solution, then AX 2 B A 1 (AX 2 ) A 1 B (A 1 A)X 2 A 1 B IX 2 X 2 A 1 B, and so A 1 B is the only solution. 33

9 Matrix inverses Remark. If n 2, because of the sizes of the matrices involved, A 1 B makes sense but BA 1 is not defined unlike the case with numbers when a 1 b ba 1 b a. Exercise 3.9 (S04 6(a)). Find the inverse of the matrix A Hence, or otherwise, find the solutions of the following systems of linear equations: (i) x + 2y 3 2y + 3z 4 x + 3y + z 0 (ii) 7x + 2y 6z 2 3x y + 3z 1 2x + y 2z 5 Exercise Solve the system y a x + 3y b. 34

10 MATRIX ALGEBRA a b d b Given a 2 2 matrix A define adj(a) :. Consider their product: c d c a a b d b ad bc 0 Aadj(A) (ad bc)i adj(a)a. c d c a 0 ad bc If deta : ad bc 0 then we can divide through by this quantity and find that: A a b deta adj(a), i.e. 1 d b. c d ad bc c a On the other hand, suppose that A is invertible. Then adj(a) I adj(a) (A 1 A)adj(A) A 1 (Aadj(A)) A 1( (deta)i ) (det A)A 1. This shows that we cannot have deta 0, since this would imply that adj(a) 0, and so A 0, which is clearly not possible since A is assumed to be invertible. Overall, we see that 2 2 matrices are invertible if and only if their determinants are nonzero. This is also true for larger matrices (once we have defined their determinants). Theorem Let A be an n n matrix and R its reduced row-echelon form. The following statements are equivalent: (i) A is invertible. (ii) The trivial solution is the only solution to AX 0. (iii) R I n. (iv) The rank of A is n. (v) There is at least one solution to AX B for every possible choice of B. (vi) There is an n n matrix C such that AC I. Proof. (i ii) If A is invertible then the unique solution of AX 0 is A 1 0 0, by Theorem 3.8. (ii iii) R I n precisely if R has a row of zeros, which is equivalent to there being more than one solution to AX 0. (iii iv) Since A is square, it is clear that R I n if and only if A has rank n (iv v) If the rank of A is equal to the number of rows then there can be no rows of zeros in the reduced row-echelon form R of A, and so the system AX B cannot be inconsistent for any B. Hence there is always a solution. On the other hand if the rank of A was less than n, then R would have a row of zeros, and we could construct a vector B for which the system would be inconsistent. (v vi) Let X 1., X 1 2.,... X 0 n.. ( ) If (v) holds then we can find n 1 vectors C 1, C 2,..., C n such that AC i X i for each i. Putting these together to make the n n matrix C [C 1 C 2 C n ] we have AC I n. On the other hand, if C exists then for any given vector B let X CB, so that AX A(CB) (AC)B IB B, i.e. there is a solution. We postpone the proof that conditions (ii vi) imply A is invertible until a later stage. 35

11 Matrix inverses This alternative characterisation of invertibility leads to a method involving row operations for calculating inverses to n n matrices. Suppose that A is invertible and write A 1 as A 1 [B 1 B 2 B n ] for column vectors B i. We want to find these B i, but since AA 1 [AB 1 AB n ] I n, we have AB i X i for the X i defined in ( ). So if we solve the n systems AB 1 X 1, AB 2 X 2,..., AB n X n then we can get the B i. This can be done by row reduction, and in fact all n systems can be solved simultaneously by applying row operations to the n 2n matrix [ A I ] and reducing A to its reduced row-echelon form R. If it turns out that R I n then A is invertible, and the matrix on the right hand side will be A 1. That is, by use of row operations, we transform to get [ A I n ] [ I n A 1 ]. Exercise Use row operations to show that A is not invertible Exercise By finding inverses, solve the system: x 2z 0 3x + y + 4z 6 2x 3y + 4z 4 36

12 MATRIX ALGEBRA Exercise Find the inverse of A Properties of matrix inverses include the following: (i) If A is invertible, so is A 1, with (A 1 ) 1 A. (ii) If A and B are invertible, so is AB, with (AB) 1 B 1 A 1. (iii) If A is invertible, so is A n for all n 1 with (A n ) 1 (A 1 ) n. (iv) If A is invertible and k 0 then ka is invertible with (ka) 1 1 k A 1. (v) If A is invertible, so is A T, with (A T ) 1 (A 1 ) T. 37

13 Matrix inverses For example if A and B are invertible then (B 1 A 1 )(AB) B 1 (A 1 A)B B 1 IB B 1 B I (AB)(B 1 A 1 ), so that (AB) 1 B 1 A 1. Exercise Find the invertible 3 3 matrix A that satisfies A Example Solving the following equation for the 2 2 matrix A: ( A T ) T [ ] A

14 MATRIX ALGEBRA ( ) T 1 A T A T (A T ) T A + 2A A 2A A A Example Find the matrix B, given that A is a another matrix such that Now A 1 A and AB B IB (A 1 A)B A 1 (AB) So if AB Example 3.18 (S05 6(b)). Find the matrix A that satisfies the equation A R1 R3; R2 R3 R4+4 R3 R2+R1 R4 4 R then R2 1 2 T so then A

15 Matrix inverses Elementary matrices An elementary matrix is any matrix that can be obtained by applying an elementary row operation to the identity matrix. For example E , E , E R 2 R 3 R 2 ( 3) R R 2 are all elementary matrices. Note what happens when multiplying any other 3 3 matrix A on the left by one of the above: E 1 A E 2 A E 3 A That is multiplying by each of the matrices E i produces the same effect on A as was used to create E i from I, and this holds true for any elementary matrix. However any row operation can be undone (swap the rows back round, or divide the row by the nonzero constant, or subtract so many copies of one row from another), and so every elementary matrix is invertible. For example, with the matrices from above, E , E , E We saw earlier that if a matrix A is invertible then its reduced row-echelon form must be I n. Suppose conversely that A can be reduced by means of row operations to I n. That means there are some elementary matrices E 1, E 2,..., E m such that E m E m 1 E 3 E 2 E 1 A I n. But each E i is invertible, hence so is the product E m E m 1 E 3 E 2 E 1. Moreover and so (E m E m 1 E 2 E 1 ) 1 (E m E m 1 E 2 E 1 )A (E m E m 1 E 2 E 1 ) 1 I n A (E m E m 1 E 2 E 1 ) 1 E 1 1 E 1 2 E 1 m 1 E 1 m. But each Ei 1 is also an elementary matrix, so A is a product of such matrices, and hence also invertible. This observation allows us to complete the proof of Theorem 3.11 (by showing (iii) (i)). Exercise Find the inverses of the following matrices, and express the matrices and their inverses as products of elementary matrices: 4 7 (i) A (ii) B

16 41 MATRIX ALGEBRA

17 Exercises Example 3.20 (S05 6(c)). Write the matrix B and its inverse as products of elementary matrices, where 2 4 B 3 7 B R2 R R1 2 R R1 1 2 [ ] R2 3 R R1 R and so B B Exercises 1. In each case solve the system of equations by finding the inverse of the coefficient matrix: (i) 2x 3y 0 x 4y 1 (ii) x + 4y + 2x 1 2x + 3y + 3z 1 4x + y + 4z 0 (iii) x + y + z + w 1 x + y 1 y + w 1 x + w Given A , find matrices B and C such that AB and CA In each case find A 1 in terms of c: (i) A 2 c c 3 (ii) A c 1 c 3 c 2 4. (a) Let A be an invertible matrix. Show that if AX AY then X Y. Show that if PA QA then P Q (b) Let A, B, C. Verify that AB CA, A is invertible, but that B C. 5. Find the inverses of the following matrices, and express the matrices and their inverses as products of elementary matrices: (i) A (ii) B (iii) C

18 MATRIX ALGEBRA 3.6 Determinants We saw that the 2 2 matrix A a b c d is invertible if and only if its determinant (det A ad bc) is nonzero. We want to extend this to all n n matrices and to do so we shall define determinants through an iterative procedure. Let A be an n n matrix, and for any 1 i, j n let A i j denote the (n 1) (n 1) matrix obtained by deleting row i and column j from A. The (i, j)-minor of A is the quantity m i j detai j ; the (i, j)-cofactor of A is c i j ( 1)i+j m i j ( 1)i+j deta i j. The factor ( 1) i+j is either +1 or 1 depending on whether or not i + j is even or odd and varies according to the following sign diagram: The signs alternate along each row and each column, with +1 in the top-left hand corner. The definition of the minors and cofactors is based on the assumption that we have defined the determinant of an (n 1) (n 1) matrix. This is the case for n 2 and n 3, since det[a] a, and the determinant of a 2 2 matrix was given earlier. Thus we can calculate m i j and ci j for matrices up to size 3 3. These numbers are then used to define/calculate the determinant of a matrix of the next size: Definition The determinant of an n n matrix A is defined to be deta : a 1 1 c1 1 + a1 2 c a1 n c1 n. That is, the entries of the top row of A are multiplied by their respective cofactors, and the results added together. Alternative notation: if A [a i j ], then a 1 1 a 1 2 a 1 n a 2 1 a 2 2 a 2 n deta a n 1 a n 2 a n n Exercise Calculate

19 Determinants Theorem For any n n matrix A and any choice of i or j deta a i 1 ci 1 + ai 2 ci ai n ci n a 1 j c1 j + a2 j c2 j + + an j cn j. That is, we can expand along any row or column, multiplying each entry by its respective cofactor, and on adding the results will always be the same. This freedom to choose which row or column to expand along can be exploited to simplify greatly any calculation. For example consider the matrix A When calculating det A expanding along the first row or along the second column gives ( 7) and ( ) (5 74 1) Expanding along the second column meant that we only had to calculate one 3 3 determinant, rather than three when expanding along the first row Exercise Calculate Definition The main diagonal of a (square) matrix A [a i j ] consists of the entries running from the top-left corner to the bottom-right corner, i.e. the numbers a 1 1, a2 2,..., an n. 44

20 MATRIX ALGEBRA (i) A is upper-triangular if the only nonzero entries are on or above the main diagonal (i.e. a i j 0 if i > j). (ii) A is lower-triangular if the only nonzero entries are on or below the main diagonal (i.e. a i j 0 if i < j). (iii) A is diagonal if the only nonzero entries are on the main diagonal. That is, if it is both upper-triangular and lower-triangular. If a matrix A is either upper-triangular or lower-triangular then its determinant is just the product of the entries on the main diagonal. For example ( 5) 30 by expanding along the first columns; expanding along the first rows gives ( 2) ( 2) ( 4) Thus it is easy to calculate determinants for triangular matrices, but not every matrix is of this form. However, any matrix that is in row-echelon form is upper-triangular, and we can transform any matrix to this form by row-operations. Alternatively when calculating determinants one can also make use of the analogous column operations. Theorem Let A and B be n n matrices. (i) If A has a row or column of zeros then det A 0. (ii) If two rows (or columns) are interchanged, the determinant of the new matrix is deta. (iii) If a row (or column) is multiplied by a constant k, the determinant of the new matrix is k deta. (iv) If a multiple of one row is added to another (or a multiple of a column added to another), the determinant of the resulting matrix is deta, i.e. unchanged. (v) det(ab) detadetb. (vi) A is invertible if and only if deta 0. (vii) deta T deta; if A is invertible then det(a 1 ) 1 deta. The second part of part (vii) follows from part (v) since if A is invertible then det(a 1 )deta det(a 1 A) deti 1. Exercise 3.27 (S04 6(b)). Compute the rank and determinant of A

21 Determinants Exercise Compute the determinant of A Exercise 3.29 (A04 6(c)). Compute Exercise For which values of x is the matrix 1 x x x 1 x invertible? x x 1 46

22 MATRIX ALGEBRA To see that invertibility of A is equivalent to deta 0 is proved in the general n n case by the same argument as in the 2 2 case, as soon as we have defined adj(a) for an n n matrix A. This is given in terms of the cofactors of the matrix A defined earlier. Indeed, the cofactor matrix of a matrix A is the matrix whose (i, j)-entry is c i j ( 1)i+j deta i j. The transpose of this matrix is the adjugate matrix (sometimes called the classical adjoint), that is c 1 1 c 2 1 c n 1 c 1 2 c 2 2 c n 2 adj(a) c 1 n c 2 n c n n a b For example in the 2 2 case if A then A c d 1 1 [d], A1 2 [c], A2 1 [b] and A2 2 [a], so that c 1 1 d, c1 2 c, c2 1 b and c2 2 a, giving T d c d b adj(a) b d c d Once we define the general n n cofactor matrix this way it is possible to show that Aadj(A) adj(a)a (deta)i, and so A is invertible if and only if deta 0 with A 1 1 det A adj(a), as before. Unfortunately as n grows large the number of operations involved in calculating A 1 this way grows far faster than if we were to use our earlier method based on row operations, so that method is usually better for computations. Example Use determinants to find which values of c make the following matrix invertible: c c 1 c 1 c c 1 c c 2 c 0 2 c 1 c 1 ( 1) 1 + c2 c 2 c [(1 + c2 )c 2c] c(1 c 2 ) where we added c R 3 to R 1, and then expanded along C 1. Thus the determinant is 0 if c 0 or c ±1, and is nonzero, hence invertible, if c is not equal to one of these three values. 47

23 Determinants However, the cofactor method does lead to another method for solving the linear system AX B, when A is invertible. This method, known as Cramer s Rule, says that the solution in this case is given by x 1 det A 1 deta, x 2 det A 2 deta,..., x n deta n det A, where A i is the matrix obtained from A by replacing column i with the column vector B. Exercise 3.32 (A03 8(b)). If A , compute the adjugate matrix adj(a) and the inverse A 1. Solve the system of equations x y + 2z 1 2x + y 3z 2 4x + y + z 0 48

24 MATRIX ALGEBRA Exercise Find x 3 where x 1 2x 2 + x 3 3 3x 1 x 3 1 4x 1 + x 2 + 2x Eigenvalues and eigenvectors Definition Let A be an n n matrix. A number λ is an eigenvalue of A if there is some nonzero column vector X 0 such that AX λx. The vector X is called an eigenvector corresponding to the eigenvalue λ. Remark. X 0 is always a solution to the equation AX λx, for any choice of λ. It is for this reason that we insist that X 0 in the definition above. On the other hand if X 0 and AX λx, i.e. if λ is an eigenvalue and X an eigenvector corresponding to this value, then for any t 0 we have tx 0 and A(tX) tax t(λx) λ(tx). That is, any nonzero multiple of X is again an eigenvector. Geometrically this says that the line through the origin in the direction of X is mapped back onto itself by the transformation Y AY. As an example, consider A and X where [ A, X AX 1 1 1] [ ] 1 [ X. 4 1] So 4 is an eigenvalue of A, and X is an eigenvector corresponding to the eigenvalue 4. On the other hand consider another vector with the same matrix A: [ [ Y AY 2 2Y. 1] 1 1][ 1 2 1] 49

25 Eigenvalues and eigenvectors So 2 is another eigenvalue, and Y is an eigenvector associated to this eigenvalue. But now note that the 2 1 vectors X and Y are not parallel i.e. they are linearly independent, and it follows that any 2 1 vector Z can be written as sx + ty for a unique choice of s, t R. For example take [ [ [ Z 3 X 3Y. 4] 1] 1] This seemingly arbitrary fact allows us to calculate simply powers of A applied to the vector Z. For example and now note that A 10 Z A 10 (X 3Y ) A 10 X + A 10 ( 3Y ) A 10 X 3A 10 Y, A 10 X A 9 (AX) 4A 9 X 4A 8 (AX) 4 2 A 8 X 4 10 X. Similarly A 10 Y ( 2) 10 Y, hence A 10 Z 4 10 [ 5 1 ] 1 3 ( 2) This is far preferable to calculating A 10, which, for the record, is A Moreover, decomposing the vector Z into a linear combination of eigenvectors can help when analysing the behaviour of the sequence of vectors AZ, A 2 Z, A 3 Z,... This prompts the question: given a matrix A how can we find its eigenvalues and corresponding eigenvectors? If λ is an eigenvector then there is some X 0 such that AX λx AX λx (A λi)x 0. That is, there is a nontrivial solution to the homogeneous system whose coefficient matrix is A λi, which happens precisely when A λi is not invertible, i.e. precisely when det(a λi) 0. The quantity det(a λi) is a polynomial in λ of degree n, called the characteristic polynomial of A. In particular it has the form det(a λi) ( 1) n λ n + b n 1 λ n b 2 λ 2 + b 1 λ + b 0 for some numbers b 0, b 1,, b n 1. The eigenvalues of A are the roots of the characteristic polynomial, and the associated eigenvectors are the nontrivial solutions of the corresponding system (A λi)x 0 which can be found by row-reduction. Example By finding the corresponding eigenvectors, show that 1, 2 and 3 are eigenvalues of the matrix Let A , then X is an eigenvector corresponding to the eigenvalue if AX X, which holds if and only if (A I)X 0. The relevant row operations on A I are R R1 2 R R1+R R R 3 R R

26 MATRIX ALGEBRA from which we get that X t 3t is the eigenvector for eigenvalue 1. t For eigenvalue 2 the calculation is (starting with A 2I): R2+4 R R1+R2; R3+R R R 1 R ( 1) and so the eigenvectors have the form X t t. 0 Finally, for the eigenvalue 3 the calculation is (starting with A 3I): R R2+4 R R3+R R R 3 R R 3 R ( 1) and so the eigenvectors have the form X 0 t. t Exercise 3.36 (S04 6(c)). Find all of the eigenvalues and eigenvectors of A

27 Eigenvalues and eigenvectors Exercise 3.37 (S03 8(c)). Let A Find all the eigenvalues of A. Find all of the eigenvectors associated to one of the eigenvalues. Example 3.38 (S05 6(d)). Find the eigenvalues of the matrix C Find the eigenvectors associated to one of the eigenvalues. 52

28 MATRIX ALGEBRA 4 λ 0 1 det(c λi) 2 1 λ λ 4 λ λ 0 2 (4 λ)(1 λ) λ λ 2 + 5λ (1 λ)( λ 2 + 5λ 6) (1 λ)(λ 2)(λ 3) and so the eigenvalues are 1, 2 and 3. For λ 1, C I R3 R R and so the components of the eigenvector satisfy x 0, z 3x 0, and y is arbitrary. That is, the eigenvectors for eigenvalue 1 are 0 t. 0 The eigenvectors for eigenvalue 2 are t 1 2 ; those for eigenvalue 3 are t Theorem Suppose that an n n matrix has n distinct eigenvalues λ 1, λ 2,...λ n, and let X 1, X 2,..., X n be corresponding eigenvectors. This set of n eigenvectors is linearly independent, hence any n 1 vector can be written uniquely as X t 1 X 1 + t 2 X t n X n. Unfortunately it is not true that every matrix has n distinct eigenvalues, but sometimes it is possible to find linearly independent eigenvectors associated to the same eigenvalue, and hope to build a set of n vectors as in the theorem above. Another potential problem is that it is possible that A has no real eigenvalues. One way to get round this limitation is to do linear algebra with complex numbers instead of real numbers. This will ensure that there is always at least one solution to det(a λi) 0, and hence at least one eigenvalue and associated eigenvector. But it may still turn out that it is impossible to find a set of n linearly independent eigenvectors. An assumption that guarantees that this will not happen is given by assuming A is symmetric: Theorem If A is an n n symmetric matrix (A A T ) with real number entries, then A has only real eigenvalues. Moreover it is possible to find a set of n linearly independent eigenvectors for A, and hence write any n 1 vector as a unique linear combination of these vectors. Exercise Find the eigenvalues and eigenvectors of the following matrices: (i) A (ii) B

29 Eigenvalues and eigenvectors Dynamical systems: predator-prey models Knowledge of eigenvalues and eigenvectors gives an insight into the behaviour of dynamical systems, such as predator-prey models which take [ place ] in discrete time. For example denote Ok the owl and rat population at time k by X k, where k is the time in months, O R k the k 54

30 MATRIX ALGEBRA number of owls, and R k the number of rats (measured in thousands). Suppose that O k+1 (0.5)O k + (0.4)R k R k+1 po k + (1.1)R k (0.5)O k represents the mortality rate of the owls; if the supply of rats is plentiful then the term (0.4)R k will be large causing the owl population to increase; (1.1)R k says that the population of rats will grow by 10%, if there were no owls, but it in fact there is a factor causing it to decrease in proportion to the population of owls according to po k, where p is some positive parameter to be specified. So we see that the change from month to month is given by Ok+1 Ok A R k+1 R k where A p 1.1 Now the eigenvalues of A are 1 10 (8 ± 9 40p). In particular if p then the eigenvalues are λ and λ , with corresponding eigenvectors 10 Y 1, Y 13 2 [ 5 1]. The eigenvalues are distinct, so any starting population X 0 can be written uniquely as X 0 t 1 Y 1 + t 2 Y 2, and then X k AX k 1 A 2 X k 2 A k X 0 t 1 A k Y 1 + t 2 A k Y 2 t 1 (1.02) k Y 1 + t 2 (0.58) k Y 2. Now as k, (0.58) k 0, and so X k t 1 (1.02) k Y 1. It follows that for large k, X k+1 t 1 (1.02) k+1 Y 1 (1.02) t 1 (1.02) k Y 1 (1.02)X k, that is, the populations of both the owls and the rats will increase at a rate of 2% each month. [ Exercise Find the eigenvalues and eigenvectors of P. Express and ] [ 0 r as sums of eigenvectors. Suppose that r, s 0, r + s 1. Determine the limit of P 1] n s as n. 55

31 Eigenvalues and eigenvectors 56

32 MATRIX ALGEBRA 3.8 Exercises 1. Compute the determinants of the following matrices: a + 1 a (i) (ii) (iii) a a (v) 0 a b a 0 c (vi) 0 a b c d (vii) b c 0 0 e x Evaluate 2 1 x 1 by first adding all other rows to the first row. 3 x Find the adjugate of the following matrices: (i) (ii) (iv) a (viii) 0 0 b p 0 c q k d s t u 4. Use determinants to find which real values c make the each of the following matrices invertible: (i) 0 c c (ii) 4 c 3 c 2 c (iii) 1 c 1 c 1 1 c c c 5 c c 5. In each case find the characteristic polynomial, eigenvalues and eigenvectors: 2 4 (i) (ii) (iii) (iv) [ Find the eigenvalues and eigenvectors of the matrix A. Write X as a ] linear combination of eigenvectors, and hence calculate A 5 X. [ Find the eigenvalues and eigenvectors of the matrix A. Write X ] as a linear combination of eigenvectors, and hence calculate A 100 X and A 201 X. 57

Linear Systems and Matrices

Linear Systems and Matrices Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.......

More information

Linear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.

Linear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0. Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the

More information

Chapter 2:Determinants. Section 2.1: Determinants by cofactor expansion

Chapter 2:Determinants. Section 2.1: Determinants by cofactor expansion Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates

More information

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1

More information

Chapter 2. Square matrices

Chapter 2. Square matrices Chapter 2. Square matrices Lecture notes for MA1111 P. Karageorgis pete@maths.tcd.ie 1/18 Invertible matrices Definition 2.1 Invertible matrices An n n matrix A is said to be invertible, if there is a

More information

Chapter 3. Determinants and Eigenvalues

Chapter 3. Determinants and Eigenvalues Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory

More information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

Equality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.

Equality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same. Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read

More information

Linear Algebra: Lecture notes from Kolman and Hill 9th edition.

Linear Algebra: Lecture notes from Kolman and Hill 9th edition. Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices

More information

4. Determinants.

4. Determinants. 4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.

More information

ECON 186 Class Notes: Linear Algebra

ECON 186 Class Notes: Linear Algebra ECON 86 Class Notes: Linear Algebra Jijian Fan Jijian Fan ECON 86 / 27 Singularity and Rank As discussed previously, squareness is a necessary condition for a matrix to be nonsingular (have an inverse).

More information

Linear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02)

Linear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02) Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 206, v 202) Contents 2 Matrices and Systems of Linear Equations 2 Systems of Linear Equations 2 Elimination, Matrix Formulation

More information

Determinants by Cofactor Expansion (III)

Determinants by Cofactor Expansion (III) Determinants by Cofactor Expansion (III) Comment: (Reminder) If A is an n n matrix, then the determinant of A can be computed as a cofactor expansion along the jth column det(a) = a1j C1j + a2j C2j +...

More information

c c c c c c c c c c a 3x3 matrix C= has a determinant determined by

c c c c c c c c c c a 3x3 matrix C= has a determinant determined by Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.

More information

Math Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p

Math Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p

More information

NOTES FOR LINEAR ALGEBRA 133

NOTES FOR LINEAR ALGEBRA 133 NOTES FOR LINEAR ALGEBRA 33 William J Anderson McGill University These are not official notes for Math 33 identical to the notes projected in class They are intended for Anderson s section 4, and are 2

More information

A FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Matrix Arithmetic

A FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Matrix Arithmetic A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Matrix Arithmetic Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

MTH 464: Computational Linear Algebra

MTH 464: Computational Linear Algebra MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)

More information

1 Matrices and Systems of Linear Equations

1 Matrices and Systems of Linear Equations Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 207, v 260) Contents Matrices and Systems of Linear Equations Systems of Linear Equations Elimination, Matrix Formulation

More information

Linear Algebra Primer

Linear Algebra Primer Introduction Linear Algebra Primer Daniel S. Stutts, Ph.D. Original Edition: 2/99 Current Edition: 4//4 This primer was written to provide a brief overview of the main concepts and methods in elementary

More information

Graduate Mathematical Economics Lecture 1

Graduate Mathematical Economics Lecture 1 Graduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 23, 2012 Outline 1 2 Course Outline ematical techniques used in graduate level economics courses Mathematics for Economists

More information

1111: Linear Algebra I

1111: Linear Algebra I 1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Michaelmas Term 2015 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Michaelmas Term 2015 1 / 10 Row expansion of the determinant Our next goal is

More information

Linear Algebra: Lecture Notes. Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway

Linear Algebra: Lecture Notes. Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway Linear Algebra: Lecture Notes Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway November 6, 23 Contents Systems of Linear Equations 2 Introduction 2 2 Elementary Row

More information

[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of

[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of . Matrices A matrix is any rectangular array of numbers. For example 3 5 6 4 8 3 3 is 3 4 matrix, i.e. a rectangular array of numbers with three rows four columns. We usually use capital letters for matrices,

More information

Elementary Linear Algebra

Elementary Linear Algebra Matrices J MUSCAT Elementary Linear Algebra Matrices Definition Dr J Muscat 2002 A matrix is a rectangular array of numbers, arranged in rows and columns a a 2 a 3 a n a 2 a 22 a 23 a 2n A = a m a mn We

More information

Introduction to Determinants

Introduction to Determinants Introduction to Determinants For any square matrix of order 2, we have found a necessary and sufficient condition for invertibility. Indeed, consider the matrix The matrix A is invertible if and only if.

More information

MAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to:

MAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to: MAC 2 Module Determinants Learning Objectives Upon completing this module, you should be able to:. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor

More information

Matrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices

Matrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices Matrices A. Fabretti Mathematics 2 A.Y. 2015/2016 Table of contents Matrix Algebra Determinant Inverse Matrix Introduction A matrix is a rectangular array of numbers. The size of a matrix is indicated

More information

Linear Algebra Primer

Linear Algebra Primer Linear Algebra Primer D.S. Stutts November 8, 995 Introduction This primer was written to provide a brief overview of the main concepts and methods in elementary linear algebra. It was not intended to

More information

Undergraduate Mathematical Economics Lecture 1

Undergraduate Mathematical Economics Lecture 1 Undergraduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 15, 2014 Outline 1 Courses Description and Requirement 2 Course Outline ematical techniques used in economics courses

More information

MATRICES AND MATRIX OPERATIONS

MATRICES AND MATRIX OPERATIONS SIZE OF THE MATRIX is defined by number of rows and columns in the matrix. For the matrix that have m rows and n columns we say the size of the matrix is m x n. If matrix have the same number of rows (n)

More information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

More information

MATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.

MATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of

More information

CS 246 Review of Linear Algebra 01/17/19

CS 246 Review of Linear Algebra 01/17/19 1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector

More information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

More information

Lecture Notes in Linear Algebra

Lecture Notes in Linear Algebra Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................

More information

a 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12

a 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12 24 8 Matrices Determinant of 2 2 matrix Given a 2 2 matrix [ ] a a A = 2 a 2 a 22 the real number a a 22 a 2 a 2 is determinant and denoted by det(a) = a a 2 a 2 a 22 Example 8 Find determinant of 2 2

More information

Chapter 7. Linear Algebra: Matrices, Vectors,

Chapter 7. Linear Algebra: Matrices, Vectors, Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.

More information

Digital Workbook for GRA 6035 Mathematics

Digital Workbook for GRA 6035 Mathematics Eivind Eriksen Digital Workbook for GRA 6035 Mathematics November 10, 2014 BI Norwegian Business School Contents Part I Lectures in GRA6035 Mathematics 1 Linear Systems and Gaussian Elimination........................

More information

1 Matrices and Systems of Linear Equations. a 1n a 2n

1 Matrices and Systems of Linear Equations. a 1n a 2n March 31, 2013 16-1 16. Systems of Linear Equations 1 Matrices and Systems of Linear Equations An m n matrix is an array A = (a ij ) of the form a 11 a 21 a m1 a 1n a 2n... a mn where each a ij is a real

More information

Math Linear Algebra Final Exam Review Sheet

Math Linear Algebra Final Exam Review Sheet Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

More information

Matrix Arithmetic. j=1

Matrix Arithmetic. j=1 An m n matrix is an array A = Matrix Arithmetic a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn of real numbers a ij An m n matrix has m rows and n columns a ij is the entry in the i-th row and j-th column

More information

and let s calculate the image of some vectors under the transformation T.

and let s calculate the image of some vectors under the transformation T. Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

More information

MATH 2030: EIGENVALUES AND EIGENVECTORS

MATH 2030: EIGENVALUES AND EIGENVECTORS MATH 2030: EIGENVALUES AND EIGENVECTORS Determinants Although we are introducing determinants in the context of matrices, the theory of determinants predates matrices by at least two hundred years Their

More information

MATH 240 Spring, Chapter 1: Linear Equations and Matrices

MATH 240 Spring, Chapter 1: Linear Equations and Matrices MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear

More information

MTH 102A - Linear Algebra II Semester

MTH 102A - Linear Algebra II Semester MTH 0A - Linear Algebra - 05-6-II Semester Arbind Kumar Lal P Field A field F is a set from which we choose our coefficients and scalars Expected properties are ) a+b and a b should be defined in it )

More information

Matrices and Linear Algebra

Matrices and Linear Algebra Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2

More information

Evaluating Determinants by Row Reduction

Evaluating Determinants by Row Reduction Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives Reduce a matrix to row echelon form and evaluate its determinant.

More information

Linear Algebra Highlights

Linear Algebra Highlights Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to

More information

Numerical Linear Algebra Homework Assignment - Week 2

Numerical Linear Algebra Homework Assignment - Week 2 Numerical Linear Algebra Homework Assignment - Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.

More information

Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations

Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2

More information

Properties of Linear Transformations from R n to R m

Properties of Linear Transformations from R n to R m Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation

More information

Linear Algebra March 16, 2019

Linear Algebra March 16, 2019 Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented

More information

Properties of the Determinant Function

Properties of the Determinant Function Properties of the Determinant Function MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Overview Today s discussion will illuminate some of the properties of the determinant:

More information

1. In this problem, if the statement is always true, circle T; otherwise, circle F.

1. In this problem, if the statement is always true, circle T; otherwise, circle F. Math 1553, Extra Practice for Midterm 3 (sections 45-65) Solutions 1 In this problem, if the statement is always true, circle T; otherwise, circle F a) T F If A is a square matrix and the homogeneous equation

More information

Chapter 2 Notes, Linear Algebra 5e Lay

Chapter 2 Notes, Linear Algebra 5e Lay Contents.1 Operations with Matrices..................................1.1 Addition and Subtraction.............................1. Multiplication by a scalar............................ 3.1.3 Multiplication

More information

Math113: Linear Algebra. Beifang Chen

Math113: Linear Algebra. Beifang Chen Math3: Linear Algebra Beifang Chen Spring 26 Contents Systems of Linear Equations 3 Systems of Linear Equations 3 Linear Systems 3 2 Geometric Interpretation 3 3 Matrices of Linear Systems 4 4 Elementary

More information

Econ Slides from Lecture 7

Econ Slides from Lecture 7 Econ 205 Sobel Econ 205 - Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for

More information

Introduction to Matrices

Introduction to Matrices 214 Analysis and Design of Feedback Control Systems Introduction to Matrices Derek Rowell October 2002 Modern system dynamics is based upon a matrix representation of the dynamic equations governing the

More information

Ma 227 Review for Systems of DEs

Ma 227 Review for Systems of DEs Ma 7 Review for Systems of DEs Matrices Basic Properties Addition and subtraction: Let A a ij mn and B b ij mn.then A B a ij b ij mn 3 A 6 B 6 4 7 6 A B 6 4 3 7 6 6 7 3 Scaler Multiplication: Let k be

More information

ENGI 9420 Lecture Notes 2 - Matrix Algebra Page Matrix operations can render the solution of a linear system much more efficient.

ENGI 9420 Lecture Notes 2 - Matrix Algebra Page Matrix operations can render the solution of a linear system much more efficient. ENGI 940 Lecture Notes - Matrix Algebra Page.0. Matrix Algebra A linear system of m equations in n unknowns, a x + a x + + a x b (where the a ij and i n n a x + a x + + a x b n n a x + a x + + a x b m

More information

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017 Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...

More information

Topic 1: Matrix diagonalization

Topic 1: Matrix diagonalization Topic : Matrix diagonalization Review of Matrices and Determinants Definition A matrix is a rectangular array of real numbers a a a m a A = a a m a n a n a nm The matrix is said to be of order n m if it

More information

OHSx XM511 Linear Algebra: Solutions to Online True/False Exercises

OHSx XM511 Linear Algebra: Solutions to Online True/False Exercises This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)

More information

Appendix A: Matrices

Appendix A: Matrices Appendix A: Matrices A matrix is a rectangular array of numbers Such arrays have rows and columns The numbers of rows and columns are referred to as the dimensions of a matrix A matrix with, say, 5 rows

More information

Lecture Summaries for Linear Algebra M51A

Lecture Summaries for Linear Algebra M51A These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture

More information

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2 Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1

More information

LINEAR ALGEBRA WITH APPLICATIONS

LINEAR ALGEBRA WITH APPLICATIONS SEVENTH EDITION LINEAR ALGEBRA WITH APPLICATIONS Instructor s Solutions Manual Steven J. Leon PREFACE This solutions manual is designed to accompany the seventh edition of Linear Algebra with Applications

More information

Recall : Eigenvalues and Eigenvectors

Recall : Eigenvalues and Eigenvectors Recall : Eigenvalues and Eigenvectors Let A be an n n matrix. If a nonzero vector x in R n satisfies Ax λx for a scalar λ, then : The scalar λ is called an eigenvalue of A. The vector x is called an eigenvector

More information

Chapter 2: Matrices and Linear Systems

Chapter 2: Matrices and Linear Systems Chapter 2: Matrices and Linear Systems Paul Pearson Outline Matrices Linear systems Row operations Inverses Determinants Matrices Definition An m n matrix A = (a ij ) is a rectangular array of real numbers

More information

Fundamentals of Engineering Analysis (650163)

Fundamentals of Engineering Analysis (650163) Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is

More information

1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued)

1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued) 1 A linear system of equations of the form Sections 75, 78 & 81 a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m can be written in matrix

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors Contents Eigenvalues and Eigenvectors. Basic Concepts. Applications of Eigenvalues and Eigenvectors 8.3 Repeated Eigenvalues and Symmetric Matrices 3.4 Numerical Determination of Eigenvalues and Eigenvectors

More information

Introduction to Matrices

Introduction to Matrices POLS 704 Introduction to Matrices Introduction to Matrices. The Cast of Characters A matrix is a rectangular array (i.e., a table) of numbers. For example, 2 3 X 4 5 6 (4 3) 7 8 9 0 0 0 Thismatrix,with4rowsand3columns,isoforder

More information

MTH Linear Algebra. Study Guide. Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education

MTH Linear Algebra. Study Guide. Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education MTH 3 Linear Algebra Study Guide Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education June 3, ii Contents Table of Contents iii Matrix Algebra. Real Life

More information

Online Exercises for Linear Algebra XM511

Online Exercises for Linear Algebra XM511 This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2

More information

Introduction. Vectors and Matrices. Vectors [1] Vectors [2]

Introduction. Vectors and Matrices. Vectors [1] Vectors [2] Introduction Vectors and Matrices Dr. TGI Fernando 1 2 Data is frequently arranged in arrays, that is, sets whose elements are indexed by one or more subscripts. Vector - one dimensional array Matrix -

More information

MAT 2037 LINEAR ALGEBRA I web:

MAT 2037 LINEAR ALGEBRA I web: MAT 237 LINEAR ALGEBRA I 2625 Dokuz Eylül University, Faculty of Science, Department of Mathematics web: Instructor: Engin Mermut http://kisideuedutr/enginmermut/ HOMEWORK 2 MATRIX ALGEBRA Textbook: Linear

More information

Fundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved

Fundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved Fundamentals of Linear Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 PREFACE Linear algebra has evolved as a branch of mathematics with wide range of applications to the natural

More information

2. Every linear system with the same number of equations as unknowns has a unique solution.

2. Every linear system with the same number of equations as unknowns has a unique solution. 1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations

More information

MTH501- Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~

MTH501- Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~ MTH501- Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~ Question No: 1 (Marks: 1) If for a linear transformation the equation T(x) =0 has only the trivial solution then T is One-to-one Onto Question

More information

M. Matrices and Linear Algebra

M. Matrices and Linear Algebra M. Matrices and Linear Algebra. Matrix algebra. In section D we calculated the determinants of square arrays of numbers. Such arrays are important in mathematics and its applications; they are called matrices.

More information

NOTES on LINEAR ALGEBRA 1

NOTES on LINEAR ALGEBRA 1 School of Economics, Management and Statistics University of Bologna Academic Year 207/8 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

More information

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column

More information

Linear Algebra Summary. Based on Linear Algebra and its applications by David C. Lay

Linear Algebra Summary. Based on Linear Algebra and its applications by David C. Lay Linear Algebra Summary Based on Linear Algebra and its applications by David C. Lay Preface The goal of this summary is to offer a complete overview of all theorems and definitions introduced in the chapters

More information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=

More information

MATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.

MATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij

More information

Study Guide for Linear Algebra Exam 2

Study Guide for Linear Algebra Exam 2 Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real

More information

MATRICES The numbers or letters in any given matrix are called its entries or elements

MATRICES The numbers or letters in any given matrix are called its entries or elements MATRICES A matrix is defined as a rectangular array of numbers. Examples are: 1 2 4 a b 1 4 5 A : B : C 0 1 3 c b 1 6 2 2 5 8 The numbers or letters in any given matrix are called its entries or elements

More information

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the

More information

2 b 3 b 4. c c 2 c 3 c 4

2 b 3 b 4. c c 2 c 3 c 4 OHSx XM511 Linear Algebra: Multiple Choice Questions for Chapter 4 a a 2 a 3 a 4 b b 1. What is the determinant of 2 b 3 b 4 c c 2 c 3 c 4? d d 2 d 3 d 4 (a) abcd (b) abcd(a b)(b c)(c d)(d a) (c) abcd(a

More information

MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants.

MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants. MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants. Elementary matrices Theorem 1 Any elementary row operation σ on matrices with n rows can be simulated as left multiplication

More information

ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices

ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS KOLMAN & HILL NOTES BY OTTO MUTZBAUER 11 Systems of Linear Equations 1 Linear Equations and Matrices Numbers in our context are either real numbers or complex

More information

LINEAR ALGEBRA REVIEW

LINEAR ALGEBRA REVIEW LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for

More information

Matrix Algebra. Matrix Algebra. Chapter 8 - S&B

Matrix Algebra. Matrix Algebra. Chapter 8 - S&B Chapter 8 - S&B Algebraic operations Matrix: The size of a matrix is indicated by the number of its rows and the number of its columns. A matrix with k rows and n columns is called a k n matrix. The number

More information

1300 Linear Algebra and Vector Geometry

1300 Linear Algebra and Vector Geometry 1300 Linear Algebra and Vector Geometry R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca May-June 2017 Matrix Inversion Algorithm One payoff from this theorem: It gives us a way to invert matrices.

More information

7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Outcomes

7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Outcomes The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a 0has a reciprocal b written as a or such that a ba = ab =. Similarly a square matrix A may have an inverse B = A where AB =

More information