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

Size: px
Start display at page:

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

Transcription

1 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) This license lets others remix, tweak, and build upon your work non-commercially, as long as they credit you and license their new creations under the identical terms. Matrix Arithmetic Page 1/80

2 Matrices - Basic Definitions and Notation Definitions Let m and n be positive integers. An m n matrix is a rectangular array of numbers having m rows and n columns. Such a matrix is said to have size m n. A row matrix (or row) is a 1 n matrix, and a column matrix (or column) is an m 1 matrix. A square matrix is an n n matrix. The (i, j)-entry of a matrix is the entry in row i and column j. For a matrix A, the (i, j)-entry of A is often written as a ij. General notation for an m n matrix, A: a 11 a 12 a a 1n a 21 a 22 a a 2n A = a 31 a 32 a a 3n.... a m1 a m2 a m3... a mn = [a ij Matrix Arithmetic Matrices Page 2/80

3 Matrices Properties and Operations 1 Equality: two matrices are equal if and only if they have the same size and the corresponding entries are equal. 2 Zero Matrix: an m n matrix with all entries equal to zero. 3 Addition: matrices must have the same size; add corresponding entries. 4 Scalar Multiplication: multiply each entry of the matrix by the scalar. 5 Negative of a Matrix: for an m n matrix A, its negative is denoted A and A = ( 1)A. 6 Subtraction: for m n matrices A and B, A B = A + ( 1)B. Matrix Arithmetic Matrices Page 3/80

4 Matrix Addition Definition Let A = [a ij and B = [b ij be two m n matrices. Then A + B = C where C is the m n matrix C = [c ij defined by c ij = a ij + b ij Example [ 1 3 Let A = 2 5 [ 0 2, B = 6 1 A + B = =. Then, [ [ Matrix Arithmetic Matrices Page 4/80

5 Theorem (Properties of Matrix Addition) Let A, B and C be m n matrices. Then the following properties hold. 1 A + B = B + A (matrix addition is commutative). 2 (A + B) + C = A + (B + C) (matrix addition is associative). 3 There exists an m n zero matrix, 0, such that A + 0 = A. (existence of an additive identity). 4 There exists an m n matrix A such that A + ( A) = 0. (existence of an additive inverse). Matrix Arithmetic Matrices Page 5/80

6 Scalar Multiplication Definition Let A = [a ij be an m n matrix and let k be a scalar. Then ka = [ka ij. Example Let A = Then A = = 3(2) 3(0) 3( 1) 3(3) 3(1) 3( 2) 3(0) 3(4) 3(5) Matrix Arithmetic Matrices Page 6/80

7 Theorem (Properties of Scalar Multiplication) Let A, B be m n matrices and let k, p R (scalars). Then the following properties hold. 1 k (A + B) = ka + kb. (scalar multiplication distributes over matrix addition). 2 (k + p) A = ka + pa. (addition distributes over scalar multiplication). 3 k (pa) = (kp) A. (scalar multiplication is associative). 4 1A = A. (existence of a multiplicative identity). Matrix Arithmetic Matrices Page 7/80

8 Example [ [ [ = [ Problem Let A and B be m n matrices. Simplify the expression 2[9(A B) + 7(2B A) 2[3(2B + A) 2(A + 3B) 5(A + B) Solution 2[9(A B) + 7(2B A) 2[3(2B + A) 2(A + 3B) 5(A + B) = 2(9A 9B + 14B 7A) 2(6B + 3A 2A 6B 5A 5B) = 2(2A + 5B) 2( 4A 5B) = 12A + 20B Matrix Arithmetic Matrices Page 8/80

9 Vectors Definitions A row matrix or column matrix is often called a vector, and such matrices are referred to as row vectors and column vectors, respectively. If X is a row vector of size 1 n, and Y is a column vector of size m 1, then we write y 1 X = [ y 2 x 1 x 2 x n and Y =. y m Matrix Arithmetic Vectors Page 9/80

10 Vector form of a system of linear equations Definition Consider the system of linear equations x 1 a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b a m1 x 1 + a m2 x a mn x n = b m Such a system can be expressed in vector form or as a vector equation by using linear combinations of column vectors: a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2. a m1 + x 2. a m2 + + x n. a mn =. b m Matrix Arithmetic Vectors Page 10/80

11 Vector form of a system of linear equations Problem Express the following system of linear equations in vector form. 2x 1 + 4x 2 3x 3 = 6 x 2 + 5x 3 = 0 x 1 + x 2 + 4x 3 = 1 Solution x x x = Matrix Arithmetic Vectors Page 11/80

12 Matrix Vector Multiplication Definition Let A = [a ij be an m n matrix with columns A 1, A 2,..., A n, written A = [ A 1 A 2 A n, and let X be an n 1 column vector, x 1 x 2 X =. Then the product of matrix A and (column) vector X is the m 1 column vector given by x 1 [ x 2 A1 A 2 A n. = x 1A 1 + x 2 A x n A n = x n that is, AX is a linear combination of the columns of A. x n n x j A j j=1 Matrix Arithmetic Matrix Vector Multiplication Page 12/80

13 Matrix Vector Multiplication Problem Compute the product AX for [ 1 4 A = 5 0 and X = [ 2 3 Solution AX = [ [ 2 3 [ 1 = 2 5 [ = [ 2 10 [ = [ Matrix Arithmetic Matrix Vector Multiplication Page 13/80

14 Matrix Vector Multiplication Problem Compute AY for A = and Y = Solution AY = ( 1) = Matrix Arithmetic Matrix Vector Multiplication Page 14/80

15 Matrix form of a system of linear equations Definition Consider the system of linear equations a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2... a m1 x 1 + a m2 x a mn x n = b m Such a system can be expressed in matrix form using matrix vector multiplication, a 11 a 12 a 1n x 1 a 21 a 22 a 2n x 2... a m1 a m2 a mn. x n = Thus a system of linear equations can be expresses as a matrix equation AX = B, where A is the coefficient matrix, B is the constant matrix, and X is the matrix of variables. b 1 b 2. b m Matrix Arithmetic Matrix Vector Multiplication Page 15/80

16 Matrix form of a system of linear equations Problem Express the following system of linear equations in matrix form. 2x 1 + 4x 2 3x 3 = 6 x 2 + 5x 3 = 0 x 1 + x 2 + 4x 3 = 1 Solution x 1 x 2 x 3 = Matrix Arithmetic Matrix Vector Multiplication Page 16/80

17 Matrix and Vector Equations Theorem 1 Every system of m linear equations in n variables can be written in the form AX = B where A is the coefficient matrix, X is the matrix of variables, and B is the constant matrix. 2 The system AX = B is consistent (i.e., has at least one solution) if and only if B is a linear combination of the columns of A. x 1 x 2 3 The vector X = is a solution to the system AX = B if and. x n only if x 1, x 2,..., x n are a solution to the vector equation x 1 A 1 + x 2 A 2 + x n A n = B where A 1, A 2,..., A n are the columns of A. Matrix Arithmetic Matrix Vector Multiplication Page 17/80

18 Problem Let A = and B = Express B as a linear combination of the columns A 1, A 2, A 3, A 4 of A, or show that this is impossible. Solution Solve the system AX = B where X is a column vector with four entries. Do so by putting the augmented matrix [ A B in reduced row-echelon form Since there are infinitely many solutions (x 4 is assigned a parameter), choose any value for x 4. Choosing x 4 = 0 (which is the simplest thing to do) gives us B = 1 1 = = A A A 3 + 0A Matrix Arithmetic Matrix Vector Multiplication Page 18/80

19 Matrix Multiplication Definition (Product of two matrices) Let A be an m n matrix and let B = [ B 1 B 2 B p be an n p matrix, whose columns are B 1, B 2,..., B p. The product of A and B is the matrix AB = A [ B 1 B 2 B p = [ AB1 AB 2 AB p i.e., the first column of AB is AB 1, the second column of AB is AB 2, etc. Note that AB has size m p. Matrix Arithmetic Matrix Multiplication Page 19/80

20 Problem Find the product AB of matrices [ A = and B = Solution AB has columns AB 1 = [ Thus, AB = [ and AB 3 = , AB 2 = [ [ , Matrix Arithmetic Matrix Multiplication Page 20/80

21 Compatibility for Matrix Multiplication Definition Let A and B be matrices, and suppose that A is m n. In order for the product AB to exist, the number of rows in B must be equal to the number of columns in A, implying that B is an n p matrix for some p. When defined, AB is an m p matrix. If the product is defined, then A and B are said to be compatible for (matrix) multiplication. Matrix Arithmetic Matrix Multiplication Page 21/80

22 Example As we saw in the previous problem [ [ = Note that the product does not exist [ Matrix Arithmetic Matrix Multiplication Page 22/80

23 Definition (The (i, j)-entry of a product) Let A = [a ij be an m n matrix and B = [b ij be an n p matrix. Then the (i, j)-entry of AB is given by Example a i1 b 1j + a i2 b 2j + + a in b nj = n a ik b kj k=1 Using the above definition, the (2, 3)-entry of the product [ is computed using the second row of the first matrix, and the third column of the second matrix, resulting in ( 1) = = 0. Matrix Arithmetic Matrix Multiplication Page 23/80

24 Matrix Multiplication is Not Commutative Problem Let A = and B = Does AB exist? If so, compute it. Does BA exist? If so, compute it. [ Solution AB = BA does not exist Matrix Arithmetic Matrix Multiplication Page 24/80

25 Problem Let G = [ 1 1 and H = [ 1 0 Does GH exist? If so, compute it. Does HG exist? If so, compute it. Solution GH = [ HG = [ 1 In this example, GH and HG both exist, but they are not equal. They aren t even the same size! Matrix Arithmetic Matrix Multiplication Page 25/80

26 Problem Let P = [ and Q = [ Solution Does PQ exist? If so, compute it. Does QP exist? If so, compute it. [ 1 1 PQ = 2 1 [ 1 1 QP = 6 3 In this example, PQ and QP both exist and are the same size, but PQ QP. Matrix Arithmetic Matrix Multiplication Page 26/80

27 Fact The three preceding problems illustrate an important property of matrix multiplication. In general, matrix multiplication is not commutative, i.e., the order of the matrices in the product is important. In other words, in general AB BA. Matrix Arithmetic Matrix Multiplication Page 27/80

28 Problem Let U = [ and V = [ Solution Does UV exist? If so, compute it. Does VU exist? If so, compute it. [ 2 4 UV = 6 8 [ 2 4 VU = 6 8 In this particular example, the matrices commute, i.e., UV = VU. Matrix Arithmetic Matrix Multiplication Page 28/80

29 Properties of Matrix Multiplication Theorem Let A, B, and C be matrices of the appropriate sizes, and let r R be a scalar. Then the following properties hold. 1 A(B + C) = AB + AC. (matrix multiplication distributes over matrix addition). 2 (B + C)A = BA + CA. (matrix multiplication distributes over matrix addition). 3 A (BC) = (AB) C. (matrix multiplication is associative). 4 r(ab) = (ra)b = A(rB). This applies to matrix-vector multiplication as well, since a vector is a row matrix or a column matrix. Matrix Arithmetic Properties of Matrix Multiplication Page 29/80

30 Elementary Proofs Problem Let A and B be m n matrices, and let C be an n p matrix. Prove that if A and B commute with C, then A + B commutes with C. Proof. We are given that AC = CA and BC = CB. Consider (A + B)C. (A + B)C = AC + BC = CA + CB = C(A + B) Since (A + B)C = C(A + B), A + B commutes with C. Matrix Arithmetic Properties of Matrix Multiplication Page 30/80

31 Problem Let A, B and C be n n matrices, and suppose that both A and B commute with C, i.e., AC = CA and BC = CB. Show that AB commutes with C. Proof. We must show that (AB)C = C(AB) given that AC = CA and BC = CB. (AB)C = A(BC) (matrix multiplication is associative) Therefore, AB commutes with C. = A(CB) (B commutes with C) = (AC)B (matrix multiplication is associative) = (CA)B (A commutes with C) = C(AB) (matrix multiplication is associative) Matrix Arithmetic Properties of Matrix Multiplication Page 31/80

32 Matrix Transposition Definition If A is an m n matrix, then its transpose, denoted A T, is the n m whose i th row is the j th column of A, 1 i n; i.e., if A = [a ij, then A T = [a ij T = [a ji i.e., the (i, j)-entry of A T is the (j, i)-entry of A. Theorem (Properties of the Transpose of a Matrix) Let A and B be m n matrices, C be a n p matrix, and r R a scalar. Then 1 (A T ) T = A 2 (ra) T = ra T 3 (A + B) T = A T + B T 4 (AC) T = C T A T Matrix Arithmetic Matrix Transposition Page 32/80

33 Problem Find the matrix A if Solution ( [ A ) T = ( [ ) T A = [ A = A = A = [ [ [ [ Matrix Arithmetic Matrix Transposition Page 33/80

34 Symmetric Matrices Definition Let A = [a ij be an m n matrix. The entries a 11, a 22, a 33,... are called the main diagonal of A. Definition The matrix A is called symmetric if and only if A T = A. Note that this immediately implies that A is a square matrix. Examples [ , , are symmetric matrices, and each is symmetric about its main diagonal. Matrix Arithmetic Matrix Transposition Page 34/80

35 Problem Show that if A and B are symmetric matrices, then A T + 2B is symmetric. Proof. (A T + 2B) T = (A T ) T + (2B) T = A + 2B T = A T + 2B, since A T = A and B T = B Since (A T + 2B) T = A T + 2B, A T + 2B is symmetric. Matrix Arithmetic Matrix Transposition Page 35/80

36 Skew Symmetric Matrices Definition An n n matrix A is said to be skew symmetric if A T = A. Example (Skew Symmetric Matrices) [ 0 2, Problem Show that if A is a square matrix, then A A T is skew-symmetric. Solution We must show that (A A T ) T = (A A T ). Using the properties of matrix addition, scalar multiplication, and transposition (A A T ) T = A T (A T ) T = A T A = (A A T ). Matrix Arithmetic Matrix Transposition Page 36/80

37 The n n Identity Matrix Definition For each n 2, the n n identity matrix, denoted I n, is the matrix having ones on its main diagonal and zeros elsewhere, and is defined for all n 2. Example I 2 = [ , I 3 = Definition Let n 2. For each j, 1 j n, we denote by E j the j th column of I n. Example When n = 3, E 1 = 1 0 0, E 2 = 0 1 0, E 3 = Matrix Arithmetic Identity Matrices and Inverses Page 37/80

38 Theorem Let A be an m n matrix Then AI n = A and I m A = A. (partial) Proof The [ (i, j)-entry of AI n is the product of the i th row of A = [a ij, namely ai1 a i2 a ij a in with the j th column of I n, namely E j. Since E j has a one in row j and zeros elsewhere, [ ai1 a i2 a ij a in Ej = a ij Instead of AI n and I m A we often write AI and IA, respectively, since the size of the identity matrix is clear from the context: the sizes of A and I must be compatible for matrix multiplication. Thus AI = A and IA = A which is why I is called an identity matrix it is an identity for matrix multiplication. Matrix Arithmetic Identity Matrices and Inverses Page 38/80

39 Matrix Inverses Definition Let A be an n n matrix. Then B is an inverse of A if and only if AB = I n and BA = I n. Note that since A and I n are both n n, B must also be an n n matrix. Example [ [ Let A = and B = AB =. Then [ and so B is an inverse of A. BA = [ Matrix Arithmetic Identity Matrices and Inverses Page 39/80

40 Not every square matrix has an inverse. Example The matrix A = [ has no inverse. To see this, suppose [ a b B = c d is an inverse of A. Then AB = which is never equal to I 2. [ [ a b c d [ c d = c d Matrix Arithmetic Identity Matrices and Inverses Page 40/80

41 Uniqueness of an Inverse Theorem If A is a square matrix and B and C are inverses of A, then B = C. Proof. Since B and C are inverses of A, AB = I = BA and AC = I = CA. Then C = CI = C(AB) = CAB and B = IB = (CA)B = CAB so B = C. Matrix Arithmetic Identity Matrices and Inverses Page 41/80

42 Example (revisited) [ [ For A = and B = 3 4 AB = [ , we saw that and BA = [ The preceding theorem tells us that B is the inverse of A, rather than just an inverse of A. Matrix Arithmetic Identity Matrices and Inverses Page 42/80

43 Definitions Let A be a square matrix, i.e., an n n matrix. The inverse of A, if it exists, is denoted A 1, and AA 1 = I = A 1 A If A has an inverse, then we say that A is invertible. Matrix Arithmetic Identity Matrices and Inverses Page 43/80

44 The inverse of a 2 2 matrix Example [ a b Suppose that A = c d. If ad bc 0, then there is a formula for A 1 : A 1 = [ 1 ad bc d c b a This can easily be verified by computing the products AA 1 and A 1 A. [ [ AA 1 a b 1 d b = c d ad bc c a ([ [ ) 1 a b d b = ad bc c d c a [ [ 1 ad bc = = ad bc 0 bc + ad 0 1 Showing that A 1 A = I 2 is left as an exercise. Matrix Arithmetic Finding the Inverse of a Matrix Page 44/80

45 The inverse of an n n matrix Problem Suppose that A is any n n matrix. How do we know whether or not A 1 exists? If A 1 exists, how do we find it? Solution The matrix inversion algorithm. Matrix Arithmetic Finding the Inverse of a Matrix Page 45/80

46 Problem [ 2 7 Let A = 5 18 Solution 1. Find the inverse of A, if it exists. Using the formula for the inverse of a 2 2 matrix: since A has an inverse and 2 ( 18) ( 7) 5 = 1 0, A 1 = 1 1 [ = Solution 2: The Matrix Inversion Algorithm [ Begin with a matrix obtained from A by augmenting A with the 2 2 identity matrix. This is written as [ [ A I = Then perform elementary row operations on this matrix until the left half of the matrix is transformed into I 2 (if possible). Matrix Arithmetic Finding the Inverse of a Matrix Page 46/80

47 Solution 2: The Matrix Inversion Algorithm (continued) [ [ [ [ [ Since it is possible to transform the left half of the matrix into I 2, the right half of the matrix is transformed into A 1. Therefore, A 1 exists, and [ A = 5 2 This is written as [ A I [ I A 1 Although the formula for the inverse of a 2 2 matrix is quicker and easier to use than the matrix inversion algorithm, the general formula for the inverse an n n matrix, n 3 (which we will see later), is more complicated and difficult to use than the matrix inversion algorithm. To find inverses of square matrices that are not 2 2, the matrix inversion algorithm is the most efficient method to use. Matrix Arithmetic Finding the Inverse of a Matrix Page 47/80

48 The Matrix Inversion Algorithm Let A be an n n matrix. To find A 1, if it exists, take the n 2n matrix [ A In obtained by augmenting A with the n n identity matrix, I n. Perform elementary row operations to transform [ A I n into a reduced row-echelon matrix. Theorem (Matrix Inverses) Let A be an n n matrix. Then the following conditions are equivalent. 1 A is invertible. 2 the reduced row-echelon form on A is I. 3 [ A In can be transformed into [ In A 1 using the Matrix Inversion Algorithm. Matrix Arithmetic Finding the Inverse of a Matrix Page 48/80

49 Problem Find, if possible, the inverse of Solution Using the matrix inversion algorithm From this, we see that A has no inverse. Matrix Arithmetic Finding the Inverse of a Matrix Page 49/80

50 Problem Let A = Find the inverse of A, if it exists. Matrix Arithmetic Finding the Inverse of a Matrix Page 50/80

51 Solution (continued) Using the matrix inversion algorithm [ A I = = [ I A Matrix Arithmetic Finding the Inverse of a Matrix Page 51/80

52 Solution (continued) Therefore, A 1 exists, and 10 A = = You can check your work by computing AA 1 and A 1 A. Matrix Arithmetic Finding the Inverse of a Matrix Page 52/80

53 Systems of Linear Equations and Inverses Suppose that a system of n linear equations in n variables is written in matrix form as AX = B, and suppose that A is invertible. Example The system of linear equations 2x 7y = 3 5x 18y = 8 can be written in matrix form as AX = B: [ [ [ 2 7 x 3 = 5 18 y 8 Previously, we found that A 1 = [ Matrix Arithmetic Finding the Inverse of a Matrix Page 53/80

54 Example (continued) Since A 1 exists and has the property that A 1 A = I, we obtain the following. AX = B A 1 (AX ) = A 1 B (A 1 A)X = A 1 B IX = A 1 B X = A 1 B i.e., AX = B has the unique solution given by X = A 1 B. Therefore, [ [ [ [ X = A = = You should verify that x = 2, y = 1 is a solution to the system. This example illustrates another method for solving a system of linear equations when the coefficient matrix is square and invertible. Unless that coefficient matrix is 2 2, this is generally NOT an efficient method for solving a system of linear equations. Matrix Arithmetic Finding the Inverse of a Matrix Page 54/80

55 Example Let A, B and C be matrices, and suppose that A is invertible. 1 If AB = AC, then A 1 (AB) = A 1 (AC) (A 1 A)B = (A 1 A)C IB = IC B = C 2 If BA = CA, then (BA)A 1 = (CA)A 1 B(AA 1 ) = C(AA 1 ) BI = CI B = C Problem Find square matrices A, B and C for which AB = AC but B C. Matrix Arithmetic Finding the Inverse of a Matrix Page 55/80

56 Inverses of Transposes and Products Example Suppose A is an invertible matrix. Then A T (A 1 ) T = (A 1 A) T = I T = I and (A 1 ) T A T = (AA 1 ) T = I T = I This means that (A T ) 1 = (A 1 ) T. Example Suppose A and B are invertible n n matrices. Then (AB)(B 1 A 1 ) = A(BB 1 )A 1 = AIA 1 = AA 1 = I and (B 1 A 1 )(AB) = B 1 (A 1 A)B = B 1 IB = B 1 B = I This means that (AB) 1 = B 1 A 1. Matrix Arithmetic Properties of the Inverse Page 56/80

57 Inverses of Transposes and Products The previous two examples prove the first two parts of the following theorem. Theorem 1 If A is an invertible matrix, then (A T ) 1 = (A 1 ) T. 2 If A and B are invertible matrices, then AB is invertible and (AB) 1 = B 1 A 1 3 If A 1, A 2,..., A k are invertible, then A 1 A 2 A k is invertible and (A 1 A 2 A k ) 1 = A 1 k A 1 k 1 A 1 2 A 1 1 (this is the obvious generalization of the inverse of a product of any number of invertible matrices) Matrix Arithmetic Properties of the Inverse Page 57/80

58 Properties of Inverses Theorem 1 I is invertible, and I 1 = I. 2 If A is invertible, so is A 1, and (A 1 ) 1 = A. 3 If A is invertible, so is A k, and (A k ) 1 = (A 1 ) k. (A k means A multiplied by itself k times) 4 If A is invertible and p R is nonzero, then pa is invertible, and (pa) 1 = 1 p A 1. Matrix Arithmetic Properties of the Inverse Page 58/80

59 Example [ 1 1 Given (3I A T ) 1 = of both sides of the equation:, we wish to find the matrix A. Taking inverses ( [ 3I A T 1 1 = = 1 [ [ 3 1 = 1 2 = ) [ Matrix Arithmetic Properties of the Inverse Page 59/80

60 Example (continued) 3I A T = A T = A T = A T = A = [ [ [ [ [ I [ Matrix Arithmetic Properties of the Inverse Page 60/80

61 Problem True or false? Justify your answer. If A 3 = 4I, then A is invertible. Solution If A 3 = 4I, then so 1 4 A3 = I ( 1 4 A2 )A = I and A( 1 4 A2 ) = I Therefore A is invertible, and A 1 = 1 4 A2. Matrix Arithmetic Properties of the Inverse Page 61/80

62 Theorem Let A be an n n matrix, and let X, B be n 1 vectors. The following conditions are equivalent. 1 A is invertible. 2 AX = 0 has only the trivial solution, X = 0. 3 A can be transformed to I n by elementary row operations. 4 The system AX = B has a unique solution X for any choice of B. 5 There exists an n n matrix C with the property that AC = I n. The following is an important and useful consequence of the above theorem. Theorem If A and B are n n matrices such that AB = I, then BA = I. Furthermore, A and B are invertible, with B = A 1 and A = B 1. Matrix Arithmetic Properties of the Inverse Page 62/80

63 Important Fact In the second Theorem, it is essential that the matrices be square. Example [ Let A = and [ AB = BA = and B = [ Then = = [ = I 2 I 3 This example illustrates why an inverse of a non-square matrix doesn t make sense. If A is m n and B is n m, where m n, then even if AB = I, it will never be the case that BA = I. Matrix Arithmetic Properties of the Inverse Page 63/80

64 Theorem If A and B are matrices such that AB = I and BA = I, then A and B are square matrices (of the same size). Matrix Arithmetic Properties of the Inverse Page 64/80

65 Elementary Matrices Definition An elementary matrix is a matrix obtained from an identity matrix by performing a single elementary row operation. The type of an elementary matrix is given by the type of row operation used to obtain the elementary matrix. Recall the elementary row operations. Elementary Row Operations. Type I: Interchange two rows. Type II: Multiply a row by a nonzero number. Type III: Add a (nonzero) multiple of one row to a different row. Matrix Arithmetic Elementary Matrices Page 65/80

66 Example E = , F = , G = are examples of elementary matrices of types I, II and III, respectively. Let 1 1 A = We are interested in the effect that (left) multiplication of A by E, F and G has on the matrix A. Computing EA, FA, and GA..., Matrix Arithmetic Elementary Matrices Page 66/80

67 Example (continued) FA = EA = GA = = = = Notice that EA is the matrix obtained from A by interchanging row 2 and row 4, which is the same row operation used to obtain E from I 4. What about FA and GA? Matrix Arithmetic Elementary Matrices Page 67/80

68 Multiplication by an Elementary Matrix Theorem Let A be an m n matrix, and suppose that B is obtained from A by performing a single elementary row operation. Then B = EA where E is the elementary matrix obtained from I m by performing the same elementary operation on I m as was performed on A. Matrix Arithmetic Elementary Matrices Page 68/80

69 Problem Let A = [ and C = [ Find elementary matrices E and F so that C = FEA. Solution Note. The statement of the problem implies that C can be obtained from A by a sequence of two elementary row operations, represented by elementary matrices E and F. [ [ [ A = 1 3 E 4 1 F = C 2 5 [ [ where E = and F =. Thus we have the sequence A EA F (EA) = C, so C = FEA, i.e., [ [ [ [ = You can check your work by doing the matrix multiplication. Matrix Arithmetic Elementary Matrices Page 69/80

70 Inverses of Elementary Matrices Example Without using the matrix inversion algorithm, find the inverse of the elementary matrix G = Hint. What row operation can be applied to G to transform it to I 4? The row operation G I 4 is to add three times row one to row three, and thus G 1 = Check by computing G 1 G. Matrix Arithmetic Elementary Matrices Page 70/80

71 Example (continued) Similarly, and E 1 = F 1 = = = Matrix Arithmetic Elementary Matrices Page 71/80

72 The Form B = UA Suppose A is an m n matrix and that B can be obtained from A by a sequence of k elementary row operations. Then there exist elementary matrices E 1, E 2,... E k such that B = E k (E k 1 ( (E 2 (E 1 A)) )) Since matrix multiplication is associative, we have B = (E k E k 1 E 2 E 1 )A or, more concisely, B = UA where U = E k E k 1 E 2 E 1. To find U so that B = UA, we could find E 1, E 2,..., E k and multiply these together (in the correct order), but there is an easier method for finding U. Matrix Arithmetic Elementary Matrices Page 72/80

73 Definition Let A be an m n matrix. We write A B if B can be obtained from A by a sequence of elementary row operations. Theorem Suppose A is an m n matrix and that A B. Then 1 there exists an invertible m m matrix U such that B = UA; 2 U can be computed by performing elementary row operations on [ A I m to transform it into [ B U ; 3 U = E k E k 1 E 2 E 1, where E 1, E 2,..., E k are elementary matrices corresponding, in order, to the elementary row operations used to obtain B from A. Matrix Arithmetic Elementary Matrices Page 73/80

74 Problem [ Let A =, and let R be the reduced row-echelon form of A Find a matrix U so that R = UA. Solution [ [ [ Starting with [ [ A I, we ve obtained R U. Therefore R = UA, where [ 1 U = [ [ Matrix Arithmetic Elementary Matrices Page 74/80

75 A Matrix as a Product of Elementary Matrices Example Let A = Suppose we do row operations to put A in reduced row-echelon form, and write down the corresponding elementary matrices E E E E E Notice that the reduced row-echelon form of A equals I 3. Now find the matrices E 1, E 2, E 3, E 4 and E 5. Matrix Arithmetic Elementary Matrices Page 75/80

76 Example (continued) E 1 = 3 1 0, E 2 = E 4 = It follows that , E 3 =, E 5 = (E 5 (E 4 (E 3 (E 2 (E 1 A))))) = I (E 5 E 4 E 3 E 2 E 1 )A = I and therefore A 1 = E 5 E 4 E 3 E 2 E 1 Matrix Arithmetic Elementary Matrices Page 76/80

77 Example (continued) Since A 1 = E 5 E 4 E 3 E 2 E 1, A 1 = E 5 E 4 E 3 E 2 E 1 (A 1 ) 1 = (E 5 E 4 E 3 E 2 E 1 ) 1 This example illustrates the following result. Theorem A = E 1 1 E 1 2 E 1 3 E 1 4 E 1 5 Let A be an n n matrix. Then, A 1 exists if and only if A can be written as the product of elementary matrices. Matrix Arithmetic Elementary Matrices Page 77/80

78 Problem [ Express A = Solution as a product of elementary matrices. [ [ E E [ 1 3 E3 0 1 [ 1 3 E4 0 1 [ 1 0 and E 1 = [ [ 1 0, E 2 = 3 1 [ 1 0, E 3 = [ 1 3, E 4 = 0 1 Since E 4 E 3 E 2 E 1 A = I, A 1 = E 4 E 3 E 2 E 1, and hence A = E 1 1 E 1 2 E 1 3 E 1 4 Matrix Arithmetic Elementary Matrices Page 78/80

79 Solution (continued) Therefore, i.e., A = [ A = 1 [ [ [ 1 [ Check your work by computing the product. [ [ [ Matrix Arithmetic Elementary Matrices Page 79/80

80 Problem Is the n n identity matrix an elementary matrix? Justify your answer. One result that we have assumed in all our work involving reduced row-echelon matrices is the following. Theorem If A is an m n matrix and R and S are reduced row-echelon forms of A, then R = S. This theorem ensures that the reduced row-echelon form of a matrix is unique, and its proof follows from the results about elementary matrices. Matrix Arithmetic Elementary Matrices Page 80/80

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

Fall Inverse of a matrix. Institute: UC San Diego. Authors: Alexander Knop

Fall Inverse of a matrix. Institute: UC San Diego. Authors: Alexander Knop Fall 2017 Inverse of a matrix Authors: Alexander Knop Institute: UC San Diego Row-Column Rule If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding

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

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

Matrix operations Linear Algebra with Computer Science Application

Matrix operations Linear Algebra with Computer Science Application Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the

More information

R n : The Cross Product

R n : The Cross Product A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler R n : The Cross Product Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)

More information

INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRICES

INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRICES 1 CHAPTER 4 MATRICES 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRICES 1 Matrices Matrices are of fundamental importance in 2-dimensional and 3-dimensional graphics programming

More information

A FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION

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

More information

Elementary Row Operations on Matrices

Elementary Row Operations on Matrices King Saud University September 17, 018 Table of contents 1 Definition A real matrix is a rectangular array whose entries are real numbers. These numbers are organized on rows and columns. An m n matrix

More information

Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes.

Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes. Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes Matrices and linear equations A matrix is an m-by-n array of numbers A = a 11 a 12 a 13 a 1n a 21 a 22 a 23 a

More information

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

1 Matrices and matrix algebra

1 Matrices and matrix algebra 1 Matrices and matrix algebra 1.1 Examples of matrices A matrix is a rectangular array of numbers and/or variables. For instance 4 2 0 3 1 A = 5 1.2 0.7 x 3 π 3 4 6 27 is a matrix with 3 rows and 5 columns

More information

3 Matrix Algebra. 3.1 Operations on matrices

3 Matrix Algebra. 3.1 Operations on matrices 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: 1 5 3 5 3 5 8

More information

Systems of Linear Equations and Matrices

Systems of Linear Equations and Matrices Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first

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

Section 9.2: Matrices.. a m1 a m2 a mn

Section 9.2: Matrices.. a m1 a m2 a mn Section 9.2: Matrices Definition: A matrix is a rectangular array of numbers: a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn In general, a ij denotes the (i, j) entry of A. That is, the entry in

More information

ICS 6N Computational Linear Algebra Matrix Algebra

ICS 6N Computational Linear Algebra Matrix Algebra ICS 6N Computational Linear Algebra Matrix Algebra Xiaohui Xie University of California, Irvine xhx@uci.edu February 2, 2017 Xiaohui Xie (UCI) ICS 6N February 2, 2017 1 / 24 Matrix Consider an m n matrix

More information

Systems of Linear Equations and Matrices

Systems of Linear Equations and Matrices Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first

More information

Elementary maths for GMT

Elementary maths for GMT Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1

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 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

Linear Algebra. The analysis of many models in the social sciences reduces to the study of systems of equations.

Linear Algebra. The analysis of many models in the social sciences reduces to the study of systems of equations. POLI 7 - Mathematical and Statistical Foundations Prof S Saiegh Fall Lecture Notes - Class 4 October 4, Linear Algebra The analysis of many models in the social sciences reduces to the study of systems

More information

Matrices. 1 a a2 1 b b 2 1 c c π

Matrices. 1 a a2 1 b b 2 1 c c π Matrices 2-3-207 A matrix is a rectangular array of numbers: 2 π 4 37 42 0 3 a a2 b b 2 c c 2 Actually, the entries can be more general than numbers, but you can think of the entries as numbers to start

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

MATRICES. a m,1 a m,n A =

MATRICES. a m,1 a m,n A = MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of

More information

Matrix Operations. Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix

Matrix Operations. Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix Matrix Operations Matrix Addition and Matrix Scalar Multiply Matrix Multiply Matrix

More information

Math 3191 Applied Linear Algebra

Math 3191 Applied Linear Algebra Math 191 Applied Linear Algebra Lecture 8: Inverse of a Matrix Stephen Billups University of Colorado at Denver Math 191Applied Linear Algebra p.1/0 Announcements We will not make it to section. tonight,

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

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

Linear Algebra and Matrix Inversion

Linear Algebra and Matrix Inversion Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much

More information

Matrix & Linear Algebra

Matrix & Linear Algebra Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A

More information

CS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages:

CS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages: CS100: DISCRETE STRUCTURES Lecture 3 Matrices Ch 3 Pages: 246-262 Matrices 2 Introduction DEFINITION 1: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n

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

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

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v 1,, v p } in n is said to be linearly independent if the vector equation x x x

More information

Matrix Arithmetic. a 11 a. A + B = + a m1 a mn. + b. a 11 + b 11 a 1n + b 1n = a m1. b m1 b mn. and scalar multiplication for matrices via.

Matrix Arithmetic. a 11 a. A + B = + a m1 a mn. + b. a 11 + b 11 a 1n + b 1n = a m1. b m1 b mn. and scalar multiplication for matrices via. Matrix Arithmetic There is an arithmetic for matrices that can be viewed as extending the arithmetic we have developed for vectors to the more general setting of rectangular arrays: if A and B are m n

More information

CLASS 12 ALGEBRA OF MATRICES

CLASS 12 ALGEBRA OF MATRICES CLASS 12 ALGEBRA OF MATRICES Deepak Sir 9811291604 SHRI SAI MASTERS TUITION CENTER CLASS 12 A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements

More information

Matrices. Chapter Definitions and Notations

Matrices. Chapter Definitions and Notations Chapter 3 Matrices 3. Definitions and Notations Matrices are yet another mathematical object. Learning about matrices means learning what they are, how they are represented, the types of operations which

More information

Phys 201. Matrices and Determinants

Phys 201. Matrices and Determinants Phys 201 Matrices and Determinants 1 1.1 Matrices 1.2 Operations of matrices 1.3 Types of matrices 1.4 Properties of matrices 1.5 Determinants 1.6 Inverse of a 3 3 matrix 2 1.1 Matrices A 2 3 7 =! " 1

More information

Math 313 Chapter 1 Review

Math 313 Chapter 1 Review Math 313 Chapter 1 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 1.1 Introduction to Systems of Linear Equations 2 2 1.2 Gaussian Elimination 3 3 1.3 Matrices and Matrix Operations

More information

Kevin James. MTHSC 3110 Section 2.1 Matrix Operations

Kevin James. MTHSC 3110 Section 2.1 Matrix Operations MTHSC 3110 Section 2.1 Matrix Operations Notation Let A be an m n matrix, that is, m rows and n columns. We ll refer to the entries of A by their row and column indices. The entry in the i th row and j

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

MATH2210 Notebook 2 Spring 2018

MATH2210 Notebook 2 Spring 2018 MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................

More information

ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,

More information

A matrix over a field F is a rectangular array of elements from F. The symbol

A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted

More information

POLI270 - Linear Algebra

POLI270 - Linear Algebra POLI7 - Linear Algebra Septemer 8th Basics a x + a x +... + a n x n b () is the linear form where a, b are parameters and x n are variables. For a given equation such as x +x you only need a variable and

More information

Determinant Worksheet Math 113

Determinant Worksheet Math 113 Determinant Worksheet Math 3 Evaluate: ) 2) 3) 4) 5) 6) 7) 8) 9) 0) ) 2) Answers ) -6 2) 9 3) - 4) 2,520 5) 0 6) 9 7) - 8) 42 9) -32 0) 64 ) 0 2) - X d2d0sl23 JK4uatfar RSFoIf0tswzaGrbeb 6LLL5CXq H 0AHl5lA

More information

MATH 2030: MATRICES. Example 0.2. Q:Define A 1 =, A. 3 4 A: We wish to find c 1, c 2, and c 3 such that. c 1 + c c

MATH 2030: MATRICES. Example 0.2. Q:Define A 1 =, A. 3 4 A: We wish to find c 1, c 2, and c 3 such that. c 1 + c c MATH 2030: MATRICES Matrix Algebra As with vectors, we may use the algebra of matrices to simplify calculations. However, matrices have operations that vectors do not possess, and so it will be of interest

More information

GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511)

GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) D. ARAPURA Gaussian elimination is the go to method for all basic linear classes including this one. We go summarize the main ideas. 1.

More information

x + 2y + 3z = 8 x + 3y = 7 x + 2z = 3

x + 2y + 3z = 8 x + 3y = 7 x + 2z = 3 Chapter 2: Solving Linear Equations 23 Elimination Using Matrices As we saw in the presentation, we can use elimination to make a system of linear equations into an upper triangular system that is easy

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

a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.

a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real

More information

Systems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University

Systems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Systems of Linear Equations By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Standard of Competency: Understanding the properties of systems of linear equations, matrices,

More information

Lecture 6 & 7. Shuanglin Shao. September 16th and 18th, 2013

Lecture 6 & 7. Shuanglin Shao. September 16th and 18th, 2013 Lecture 6 & 7 Shuanglin Shao September 16th and 18th, 2013 1 Elementary matrices 2 Equivalence Theorem 3 A method of inverting matrices Def An n n matrice is called an elementary matrix if it can be obtained

More information

Section 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns.

Section 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. Section 9.2: Matrices Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. That is, a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn A

More information

MAC Module 2 Systems of Linear Equations and Matrices II. Learning Objectives. Upon completing this module, you should be able to :

MAC Module 2 Systems of Linear Equations and Matrices II. Learning Objectives. Upon completing this module, you should be able to : MAC 0 Module Systems of Linear Equations and Matrices II Learning Objectives Upon completing this module, you should be able to :. Find the inverse of a square matrix.. Determine whether a matrix is invertible..

More information

Matrices and systems of linear equations

Matrices and systems of linear equations Matrices and systems of linear equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T.

More information

Topics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij

Topics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij Topics Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij or a ij lives in row i and column j Definition of a matrix

More information

Chapter 2. Matrix Arithmetic. Chapter 2

Chapter 2. Matrix Arithmetic. Chapter 2 Matrix Arithmetic Matrix Addition and Subtraction Addition and subtraction act element-wise on matrices. In order for the addition/subtraction (A B) to be possible, the two matrices A and B must have the

More information

Review Let A, B, and C be matrices of the same size, and let r and s be scalars. Then

Review Let A, B, and C be matrices of the same size, and let r and s be scalars. Then 1 Sec 21 Matrix Operations Review Let A, B, and C be matrices of the same size, and let r and s be scalars Then (i) A + B = B + A (iv) r(a + B) = ra + rb (ii) (A + B) + C = A + (B + C) (v) (r + s)a = ra

More information

is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ]

is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ] Matrices: Definition: An m n matrix, A m n is a rectangular array of numbers with m rows and n columns: a, a, a,n a, a, a,n A m,n =...... a m, a m, a m,n Each a i,j is the entry at the i th row, j th column.

More information

Applied Matrix Algebra Lecture Notes Section 2.2. Gerald Höhn Department of Mathematics, Kansas State University

Applied Matrix Algebra Lecture Notes Section 2.2. Gerald Höhn Department of Mathematics, Kansas State University Applied Matrix Algebra Lecture Notes Section 22 Gerald Höhn Department of Mathematics, Kansas State University September, 216 Chapter 2 Matrices 22 Inverses Let (S) a 11 x 1 + a 12 x 2 + +a 1n x n = b

More information

Prepared by: M. S. KumarSwamy, TGT(Maths) Page

Prepared by: M. S. KumarSwamy, TGT(Maths) Page Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 50 - CHAPTER 3: MATRICES QUICK REVISION (Important Concepts & Formulae) MARKS WEIGHTAGE 03 marks Matrix A matrix is an ordered rectangular array of numbers

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

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J Olver 3 Review of Matrix Algebra Vectors and matrices are essential for modern analysis of systems of equations algebrai, differential, functional, etc In this

More information

Inverses and Elementary Matrices

Inverses and Elementary Matrices Inverses and Elementary Matrices 1-12-2013 Matrix inversion gives a method for solving some systems of equations Suppose 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 n1 x

More information

MATH10212 Linear Algebra B Homework Week 5

MATH10212 Linear Algebra B Homework Week 5 MATH Linear Algebra B Homework Week 5 Students are strongly advised to acquire a copy of the Textbook: D C Lay Linear Algebra its Applications Pearson 6 (or other editions) Normally homework assignments

More information

Elementary Linear Algebra

Elementary Linear Algebra Elementary Linear Algebra Linear algebra is the study of; linear sets of equations and their transformation properties. Linear algebra allows the analysis of; rotations in space, least squares fitting,

More information

Inverting Matrices. 1 Properties of Transpose. 2 Matrix Algebra. P. Danziger 3.2, 3.3

Inverting Matrices. 1 Properties of Transpose. 2 Matrix Algebra. P. Danziger 3.2, 3.3 3., 3.3 Inverting Matrices P. Danziger 1 Properties of Transpose Transpose has higher precedence than multiplication and addition, so AB T A ( B T and A + B T A + ( B T As opposed to the bracketed expressions

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

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

MATH Mathematics for Agriculture II

MATH Mathematics for Agriculture II MATH 10240 Mathematics for Agriculture II Academic year 2018 2019 UCD School of Mathematics and Statistics Contents Chapter 1. Linear Algebra 1 1. Introduction to Matrices 1 2. Matrix Multiplication 3

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

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

n n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full

n n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full n n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in n variables x 1, x 2,..., x n a 11 x 1 + a 12 x

More information

Lectures on Linear Algebra for IT

Lectures on Linear Algebra for IT Lectures on Linear Algebra for IT by Mgr Tereza Kovářová, PhD following content of lectures by Ing Petr Beremlijski, PhD Department of Applied Mathematics, VSB - TU Ostrava Czech Republic 3 Inverse Matrix

More information

Matrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices

Matrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices Graphics 2009/2010, period 1 Lecture 4 Matrices m n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in

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

Math Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT

Math Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear

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

Matrix representation of a linear map

Matrix representation of a linear map Matrix representation of a linear map As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors

More information

Linear Algebra. Linear Equations and Matrices. Copyright 2005, W.R. Winfrey

Linear Algebra. Linear Equations and Matrices. Copyright 2005, W.R. Winfrey Copyright 2005, W.R. Winfrey Topics Preliminaries Systems of Linear Equations Matrices Algebraic Properties of Matrix Operations Special Types of Matrices and Partitioned Matrices Matrix Transformations

More information

Math 4377/6308 Advanced Linear Algebra

Math 4377/6308 Advanced Linear Algebra 2.3 Composition Math 4377/6308 Advanced Linear Algebra 2.3 Composition of Linear Transformations Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math4377

More information

Linear Algebra V = T = ( 4 3 ).

Linear Algebra V = T = ( 4 3 ). Linear Algebra Vectors A column vector is a list of numbers stored vertically The dimension of a column vector is the number of values in the vector W is a -dimensional column vector and V is a 5-dimensional

More information

Matrices BUSINESS MATHEMATICS

Matrices BUSINESS MATHEMATICS Matrices BUSINESS MATHEMATICS 1 CONTENTS Matrices Special matrices Operations with matrices Matrix multipication More operations with matrices Matrix transposition Symmetric matrices Old exam question

More information

A primer on matrices

A primer on matrices A primer on matrices Stephen Boyd August 4, 2007 These notes describe the notation of matrices, the mechanics of matrix manipulation, and how to use matrices to formulate and solve sets of simultaneous

More information

ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Corrected Version, 7th April 013 Comments to the author at keithmatt@gmail.com Chapter 1 LINEAR EQUATIONS 1.1

More information

Solving Linear Systems Using Gaussian Elimination

Solving Linear Systems Using Gaussian Elimination Solving Linear Systems Using Gaussian Elimination DEFINITION: A linear equation in the variables x 1,..., x n is an equation that can be written in the form a 1 x 1 +...+a n x n = b, where a 1,...,a n

More information

Section 12.4 Algebra of Matrices

Section 12.4 Algebra of Matrices 244 Section 2.4 Algebra of Matrices Before we can discuss Matrix Algebra, we need to have a clear idea what it means to say that two matrices are equal. Let's start a definition. Equal Matrices Two matrices

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

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

What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010

What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010 What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010 Scott Strong sstrong@mines.edu Colorado School of Mines What is the Matrix? p. 1/20 Overview/Keywords/References Advanced

More information

Matrices and RRE Form

Matrices and RRE Form Matrices and RRE Form Notation R is the real numbers, C is the complex numbers (we will only consider complex numbers towards the end of the course) is read as an element of For instance, x R means that

More information

Calculus II - Basic Matrix Operations

Calculus II - Basic Matrix Operations Calculus II - Basic Matrix Operations Ryan C Daileda Terminology A matrix is a rectangular array of numbers, for example 7,, 7 7 9, or / / /4 / / /4 / / /4 / /6 The numbers in any matrix are called its

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

DM559 Linear and Integer Programming. Lecture 3 Matrix Operations. Marco Chiarandini

DM559 Linear and Integer Programming. Lecture 3 Matrix Operations. Marco Chiarandini DM559 Linear and Integer Programming Lecture 3 Matrix Operations Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline and 1 2 3 and 4 2 Outline and 1 2

More information

Math 360 Linear Algebra Fall Class Notes. a a a a a a. a a a

Math 360 Linear Algebra Fall Class Notes. a a a a a a. a a a Math 360 Linear Algebra Fall 2008 9-10-08 Class Notes Matrices As we have already seen, a matrix is a rectangular array of numbers. If a matrix A has m columns and n rows, we say that its dimensions are

More information

Review : Powers of a matrix

Review : Powers of a matrix Review : Powers of a matrix Given a square matrix A and a positive integer k, we define A k = AA A } {{ } k times Note that the multiplications AA, AAA,... make sense. Example. Suppose A=. Then A 0 2 =

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION,, 1 n A linear equation in the variables equation that can be written in the form a a a b 1 1 2 2 n n a a is an where

More information

Linear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds

Linear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds Linear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds These notes are meant to provide a brief introduction to the topics from Linear Algebra that will be useful in Math3315/CSE3365, Introduction

More information