Matrices and Determinants


 Lilian Shaw
 1 years ago
 Views:
Transcription
1 Chapter1 Matrices and Determinants 11 INTRODUCTION Matrix means an arrangement or array Matrices (plural of matrix) were introduced by Cayley in 1860 A matrix A is rectangular array of m n numbers (or functions) arranged in m horizontal lines (known as rows ) and in n vertical lines (known as columns ), denoted by A m n These m n numbers are known as the elements or entries of the matrix A and are enclosed in brackets [ ], or()or Theorder of the matrix is m n When m n, the matrix is said to be rectangular Row matrix (or row vector) B 1 n is a matrix having only one row (and several columns), column matrix (or column vector) C m 1 is a matrix having only one column (and several rows) A matrix is said to be a nsquare matrix or simply square matrix if m n Thus the number of rows and number of columns in a square matrix are equal The elements of the matrix A are denoted by a ij and are located by the double subscript notation ij where the first subscript i denotes the row (position) and the second subscript j denotes the column position Thus capital letters are used to denote matrices, while the corresponding small letters with double subscript notation are used to denote the elements (or entries) Thus A [a ij ] Null or zero matrix denoted by 0 is a matrix with all its elements zero Equality: Two matrices A and B are said to be equal if they are of the same order and a ij b ij for every i and j Otherwise they are unequal, denoted by A B 12 MATRIX ALGEBRA Sum (or difference): C m n A m n ± B m n then C is said to be the sum (or difference) of A and B provided c ij a ij ± b ij for i, j ie the elements of C are obtained by adding (or subtracting) the corresponding elements of A and B Note that addition or subtraction of matrices A and B is possible only when both A and B are of the same order Submatrix of A is a matrix obtained from A after deleting some rows or columns or both Scalar multiplication: For any non zero scalar k, we have C ka when c ij ka ij ie every element of A is multiplied by k Thus B is considered as B multiplied by 1 Properties: 1 A + B B + A commutative 2 A + (B ± C) (A + B) ± C associative 3 k(a + B) ka + kb distributive 4 A B B A not commutative Transpose of a matrix A of order m n is denoted by A T or A is obtained from A by interchanging the rows and columns Thus B A T is of n m order and b ji a ij for any i, j, ie the i, j th element of A is placed in the j, i th location in A T Properties: 1 (A T ) T A 2 (ka) T ka T 3 (A + B) T A T + B T 11
2 12 MATHEMATICAL METHODS Matrix multiplication: Two matrices A and B are said to be conformable for multiplication if the number of columns in A is equal to the number of rows in B Then the product of two matrices A m p and B p n is a matrix c m n where c ij p a ik b kj k1 for i 1tom and j 1ton ie, i, j th element in the product matrix C is obtained by adding the p products obtained by multiplying each entry of the i th row of A by the corresponding entry of the j th column of B Thus matrix multiplication amounts to multiplication of rows (of the first matrix) into columns (of the second matrix) ie, row by column multiplication p n m A p B m C n ie mn scalar products of the m rows of A with n columns of B In the product C AB, the matrix B is premultiplied or multiplied from the left by A; while the matrix A is post multiplied or multiplied from the right by B Properties: 1 (ka)b k(ab) A(kB) kab 2 A(BC) (AB)C 3 (A + B)C AC + BC 4 A(B + C) AB + AC However 5 AB BA in general (not commutative) 6 AB 0 does not necessarily imply that A 0or B 0orBA 0 Also 7 AB AC does not necessarily imply that B C, even when A 0 8 (AB) T B T A T ie, transpose of a product is the product of the transposes 13 SPECIAL SQUARE MATRICES The elements a ii of an nsquare matrix are known as diagonal elements Trace of matrix A trace of A n a ii sum of the diagonal elements i1 Result: trace (A + B) trace A + trace B, trace (ka) k trace A A is singular matrix if A 0 A is Nonsingular matrix if A 0 Upper triangular matrix if a ij 0fori>jie, can have non zero entries only on and above the main diagonal while any entry below the diagonal is zero Lower triangular matrix if a ij 0fori<jA matrix is said to be triangular if it is either upper or lower triangular matrix Diagonal matrix if any entry above or below, the main diagonal is zero However zero entries may be present in the diagonal Thus a ij 0fori j (however a ii 0, not for all i) Scalar matrix is a diagonal matrix in which all the diagonal entries are equal to a constant k ie, a ii k for every i and a ij 0 for any i and j, (i j) Identity matrix denoted by I is a scalar matrix with k 1 Thus I 3 I Positive integral power of a matrix A, denoted by A n, is obtained by multiplying A by itself n times 14 DIFFERENCES BETWEEN DETERMINANTS AND MATRICES Determinant D Matrix A 1 It has a numerical value It has no value It is a symbol representing an array of many numbers on which algebra can be performed 2 It can only a be square It can be rectangular 3 It is zero when elements It is zero only when all the of any one row (or elements in the matrix are zero column) are zero 4 It is multiplied by k It is multiplied by k if all the if elements of any elements of the matrix are one row (or column) multiplied by k are multiplied by k 5 Its value remains unalt It gets altered (giving rise to ered by the interchange a new matrix) when rows and of rows and columns columns are interchanged 6 Its value is D when It gets changed to a new matrix adjacent rows (or when adjacent rows (or columns) are interchanged columns) are interchanged
3 MATRICES AND DETERMINANTS 13 WORKED OUT EXAMPLES Example 1: Classify the following matrices Also find the order of the matrices (a) (b) 2 3 (c)[ ] (d) (e) k (f) (g) 0 k k k (h) (i) [13] Solution: (a) Rectangular matrix of order 5 4 (b) Column matrix of order 4 1 (c) Row matrix of order 1 5 (d) Upper triangular matrix, 4 4 (square) (e) Lower triangular matrix, 3 3 (square) (f) Diagonal matrix, 3 3 (square) (g) Scalar matrix, 4 4 (square) (h) Null or zero matrix, 3 3 (square) (i) 1 1 matrix which identifies the single entry Example 2: Suppose matrix A has m rows and m + 6 columns and matrix B has n rows and 12 n columns If both AB and BA exists, determine the orders of the matrices A and B Solution: Since A m m + 6 B n 12 n exists, the number of columns in A must be equal to the number of rows in B ie m +6nor m n 6 Similarly B n 12 n A m m + 6 exists, 12 n m or m + n 12 solving m 3,n 9 The order of A is 3 9 and of B is 9 3 Example [ ] 3: Determine [ AB ] and BA if A B and A + B IsAB BA Solution: Adding A B [ ] 6 7 A + B D we get 5 2 [ ] 2 3 C 1 2 [ ] 8 10 so 6 4 [ ] [ ] A C + D [ ] 4 5 A Then 3 2 [ ] [ ] [ ] B D A [ ][ ] [ ] So AB and [ ][ ] [ ] BA Note that AB BA, in general [ ] Example 4: If A show that A 4 9 n [ ] n 25n using mathematical induction 4n 1 10n Solution: Consider A 2 A A [ ][ ] [ ][ ] A [ ][ ] Now A 3 A A [ ] So [ ] A [ ] k 25 k Assume A k Then 4 k 1 10 k
4 14 MATHEMATICAL METHODS [ ][ ] A k+1 A A k k 25k 4 9 4k 1 10k [ ] k 25 25k 4 + 4k 9 10k [ ] (k + 1) 25(k + 1) 4(k + 1) 1 10(k + 1) By mathematical induction the result follows ( ) 8 4 Example 5: If A prove that 2 2 A 2 10 A + 24 I 0 ( )( ) Solution: A 2 A A ( ) Then A A + 24I ( ) ( ) ( ) ( ) ( ) Example 6: Show that AB AC does not necessarily imply that B C where A 2 1 3, B , C Solution: A 3 3 B 3 4 D A 3 3 C 3 4 however B C Example 7: Verify that (a) AB BA 0, (b) AC A, (c) CA C, (d) ACB CBA, (e) (A ± B) 2 A 2 + B 2, (f) (A B)(A + B)A 2 B where A 1 4 5, B 1 3 5, C Solution: (a) AB Similarly BA Thus AB BA (b) AC A (c) CA C (d) ACB (AC)B (A)B AB 0 CBA C(BA)C 00 (e) (A ± B) 2 A 2 ± AB ± BA + B 2 A 2 + B 2 since AB BA 0 (f) (A B)(A + B) A 2 BA + AB B 2 A 2 B 2 since AB BA 0 Example 8: Prove that (a) trace (A + B) trace A + trace B (b) trace (ka) k trace A n Solution: (a) Trace (A + B) (a ii + b ii ) a ii + b ii trace A + trace B i1
5 (b) Trace (ka) n (ka ii )k n a ii k trace A i1 i Example 9: Express A as the product of LU where L and U are lower and upper triangular matrices (known as LUdecomposition or Factorization) Solution: Let the lower triangular matrix be l u 11 u 12 u 13 L l 21 l 22 0 while U 0 u 22 u 23 l 31 l 32 l u 33 be the upper triangular matrix Then A LU l u 11 u 12 u 13 l 21 l u 22 u 23 l 31 l 32 l u 33 Equating the corresponding component on both sides, we have l 11 u 11 5,l 11 u 12 2,l 11 u 13 1, l 21 u 11 7,l 21 u 12 + l 22 u 22 1,l 21 u 13 + l 22 u 23 5, l 31 u 11 3,l 31 u 12 + l 32 u 22 7,l 31 u 13 + l 32 u 23 +l 33 u 33 4 Since there are 12 unknowns and 9 equations only, to get a unique solution, assume that l 11 l 22 l 33 1 Now u 11 5,u 12 2,u 13 1 Then l 21 7 u , u 22 (1 l 21 u 12 )/l 22 [ ( 2)]/ , u 23 ( 5 l 21 u 13 )/l 22 ( ) 32 5,l 31 3 u Similarly l , u Thus A LU MATRICES EXERCISE If A 5 0 2, B 4 2 5, MATRICES AND DETERMINANTS C find (a) A + B, (b)a B, (c) 3B,(d) verify A + (B C) (A + B) C, (e) Find D such that C + D B, (f) AB, (g)ba, (h)isab BA, (i) AC, (j) verify A(B + C) AB + AC (k) Is AB AC, (l) Is AC CA Ans (a) A + B (b) A B (c) 3B (e) Hint: B C D (f) AB (g) BA (h) No In general AB BA (i) AC (j) Hint: B + C (l) No (j) Hint: CA , No AC CA 2 Determine the orders of the matrices A having m rows and m + 5 columns and B having n rows and 11 n columns if both AB and BA exist
6 16 MATHEMATICAL METHODS Ans A 3 8 ; B Compute [ the ] product AB, [ BA given ] that A + B and A B IsBA AB [ ] [ ] Ans AB, BA,No [ ] [ ] Hint: A, B State why in general (a) (A ± B) 2 A 2 ± 2AB + B 2 (b) A 2 B 2 (A B)(A + B) (c) Verify the results (a) and (b) for A and B in the above problem 3 Ans Since AB BA in general If A 0 2 1, verify that A 3 5A A 4I 0 [ ] cos θ sin θ 6 If A show that A sin θ cos θ [ ] cos nθ sin nθ by using mathematical sin nθ cos nθ induction 7 Is AB BA given that A 1 2 1, B Determine the values of x for which the matrix A 1 2x 3x is nonsingular where A x 3 0 Ans A is nonsingular for any x other than 3 and 2 3 Hint: A 3x 2 11x+6 (x 3) ( x 2 3) 0 [ ] [ ] If A, B then find (a) A T (b) B T (c) (A + B) T (d) (A B) T (e) A T + B T (f) A T B T (g) Verify (A + B) T A T + B T (h)is(a B) T A T B T (i) (AB) T (j) B T A T (k) Verify that (AB) T B T A T [ ] [ ] Ans (a) A T (b) B 1 1 T 5 2 [ ] 4 7 (c) (A + B) T 6 3 [ ] 2 7 (d) (A B) T 4 1 [ ] 4 7 (e) A T + B T (g) True 6 3 (h) (A B) T A T B T [ ] 10 7 (i) (AB) T 17 2 [ ] 10 7 (j) B T A T 17 2 (k) True Express A as product LU where L and U are lower and upper triangular matrices Ans L 0 1 0, U DETERMINANTS Although determinants are inefficient in practical computations, they are useful in vector algebra, differential equations and eigenvalue problems A determinant is a scalar (numerical value) associated with only square matrix A [a ij ] and is denoted as determinant of A or det A or A Thus a determinant is a scalarvalued function whose domain is a set of square matrices A determinant of an n n square matrix A is a scalar given by a 11 a 12 a 1n a 21 a 22 a 2n D det A (1) a n1 a n2 a nn The determinant D issaidtobeofordern and contains n 2 elements or quantities (which may be numbers or functions), arranged in n rows and n columns
7 MATRICES AND DETERMINANTS 17 The principal diagonal of the determinant is the sloping line of elements from left top corner a 11 to a nn Note that in the matrix representation the elements a ij are enclosed between brackets []or()or, whereas in the determinant the elements are enclosed between vertical lines or bars For n 2, the second order determinant is defined by D deta a 11 a 12 a 21 a 2 a 11 a 22 a 21 a 12 (2) ie second order determinant difference between the product of elements of principal diagonal and the product of the elements of the other diagonal For n 1, D det[a 11 ] [a 11 ] a 11 Note: Here vertical bars does not denote absolute value Thus det [ 5] 5 5 Minor of an element a ij of a matrix A, denoted by M ij,isan(n 1) order determinant of the submatrix of A obtained by omitting the ith row and jth column in A Cofactor of an element a ij of a matrix A, denoted by C ij, is a signed minor of a ij ie C ij ( 1) i+j M ij LaplaceExpansion Laplace Expansion is the expansion of determinant in terms of the cofactors For n 2 n D a ij C ij a i1 C i1 + a i2 C i2 + +a in C in j1 (row wise for any i 1, 2, or n) (3) or D n a ij C ij a 1j C 1j +a 2j C 2j + +a nj C nj (4) i1 (column wise for any j 1, 2,or n) Thus the value of a determinant is the sum of the products of elements of any row (or column) and their respective cofactors However sum of products formed by multiplying the elements of a row (or column) of A by the corresponding cofactors of another n row (or column) of A is zero ie a ik C jk δ ij A k1 n or a kj C ki δ ij A Here δ ij is the Kronecker k1 delta So it is convenient to choose the row or column in the determinant with zeros in it, since these terms in the expansion will vanish The expansion of D by (3) or (4) involves n! determinants since D is defined in terms of n determinants of order (n 1), each of which is in turn defined in terms of (n 1) determinants of order (n 2) and then (n 2) determinants of order (n 3) and so on As the number of calculations of an nth order determinant is N(n) en!, even for n 25, computing time is sec s years However with the use of several properties of determinants which are listed below, the determinant can be triangularized In this case, the number of calculations N(n) 2n3 Forn 25, computation time 3 is 001 second (against years which is incredible!) Properties of Determinants 1 If all the elements of any one row (or column) are zero, then det A 0 2 If any two rows (or columns) are proportional to each other, then det A 0 3 If any row (or column) is a linear combination of other rows (or columns), then det A 0 4 If all the elements of any row or column are multiplied by k, giving rise to a new matrix B, then det B k det A 5 If B ka then det B det (ka) k n det A 6 det (A T ) det (A) 7 In general, det (αa + βb) α det (A) + β det (B) iedeterminant()isnot linear 8 If any two rows (or columns) of A are interchanged, yielding a new matrix B then det B det A 9 If k times the elements of any row (or column) in A are added to the corresponding elements of any row (or column) in A, giving rise to a new matrix B, then det B det A This operation is written symbolically as
8 18 MATHEMATICAL METHODS r i r i + kr j ie, elements of jth row multiplied by k are added to the corresponding elements of theith row Here the ith row gets modified Similarly c i c i + kc j 10 If A is a (upper or lower) triangular matrix or diagonal matrix then det A a 11 a 22 a nn ie value of the determinant is the product of the diagonal elements 11 If each element of a row (or column) consists of m terms (two: binomials, three: trinomial etc), then the determinant is expressed as the sum of m determinants Suppose any row (or column) of A is a binomial say a b + c then det A a det A b + det A c Here A a is the original matrix, A b is the matrix obtained from the original matrix A a by replacing a by b and similarly A c by replacing a by c Extending this, if the elements of say three rows (or columns) consists of m, n, p terms respectively then the original determinants can be expressed as the sum of m n p determinants as stated above Product of Determinants 12 For any n n matrices A and B det (AB) det (BA) det A det B Note: If A is singular, then AB is also singular so det A 0, det AB 0Thus00 13 If C det A det B, then the i, jth element of C is obtained by multiplying the ith row (or column) of A with jth column (or row) of B Note that in matrix multiplication i, j th element is obtained by multiplying the ith row of A with jth column of B Whereas in determinant multiplication, ij th element is obtained either by multiplying (ith row of A with jth column of B)or(ith row of A with jth row of B)or(ith column of A with jth column of B)or(ith column of A with jth row of B) Thus, in determinant multiplication we can multiply row by row, row by column, column by row or column by column 14 Derivative of a Determinant If the elements a ij of a matrix A are differentiable functions of a parameter t, then the derivative of the determinant A equals to sum of n determinants obtained by replacing in all possible ways the elements of one row (or column) of A by their derivatives wrt t, ie da 11 da 1n dt dt a 11 a 1n d a 21 a 2n da 21 da 2n dt dt (det A) dt + + a n1 a nn a n1 a nn a 11 a 1n + + a n 1, 1 a n+1,n da n1 dt da nn dt 15 Factor Theorem Consider an nth order functional determinant A in which the elements a ij are functions of x Suppose for x x, any two rows (or columns) of A become equal, then det A 0 Then det A must contain a factor (x x ) Suppose for x x,k rows (or columns) become identical then det A 0 So consequently det A must contain a factor (x x ) k 1 16 Singular A matrix A is said to be singular if A 0, otherwise A is said to be nonsingular ie, determinant of A is non zero ( A 0) WORKED OUT EXAMPLES Example 1: Find all the cofactors and evaluate A Solution: C 11 cofactor of 2 5, C 12 cofactor of 3 4, C 21 cofactor of 4 3,
9 MATRICES AND DETERMINANTS 19 C 22 cofactor of 5 2 By Laplace expansion about the first row, A 2 C C 12 2(5) + 3(4) By Laplace expansion about the second row A 4 C C 22 4( 3) + 5( 2) Similarly about first column A ( 2)(5) 4( 3) About 2nd column, A 3(+4) + 5( 2) 2 Example 2: Find the minors M 21,M 13, cofactors C 22,C 32 and evaluate the determinant A Solution: M 21 minor of M 13 Minor of C 22 cofactor of 18 ( 1) 2+2 M C 32 cofactor of 15 ( 1) 3+2 M ( 48) 48 Expanding the determinant by element of first row, we have A 12( ) 27( ) +12( ) 12(360) 27( 560) + 12( 840) Example 3: where Evaluate A by triangularization A Solution: R 2 R 2 2 R 1,R 3 R 3 3 R 1, R 4 R 4 4 R A Expanding by first column and taking minus from the three rows, A ( 1) R 2 R 2 R ( 1) R 1 R 1 + 3R ( 1) R 3 R 3 + 5R Expanding by first column A ( 1) ( 1)( 1)[ ] A [ ] 80 Example 4: Evaluate A Solution: Interchanging all the rows and columns we get A B ( ) since B is an lower triangular matrix Example 5: Find the value of
10 110 MATHEMATICAL METHODS Solution: R 3 R 3 R 2,R 4 R 4 R Taking 4 from 3rd row and 12 from 4th row, ; R 4 R 4 + R since all the entries of the 4th row are zero Example 6: a + 2b a + 4b a + 6b Evaluate a + 3b a + 5b a + 7b a + 4b a + 6b a + 8b Solution: Performing R 3 R 3 R 2,R 1 R 1 R 2, a + 2b a + 4b a + 6b b b b 2b 2b 2b 0sincethelasttworow are proportional Example 7: Solution: column, Evaluate the nth order determinant a b b b b a b b b b a b b b b a Adding all the (n 1) columns to the first a + (n 1)b b b a + (n 1)b a b a + (n 1)b b b a + (n 1)b b a 1 b b b 1 a b b a + (n 1)b 1 b a b 1 b b a when a b, R 1 R 2 R 3 R n ie all the n rows are identical and so determinant vanish Thus by factor theorem, (a b) n 1 is factor of Thus (a b) n 1 [a + (n 1)b] a + b b+ c c+ a Example 8: Show that b + c c+ a a + b c + a a + b b+ c a b c 2 c a b Solution: Here the 3 rows contains binomials So the given determinant can be expressed as the sum of determinants as follows a + b b+ c c+ a b + c c+ a a + b c + a a + b b+ c a b+ c c+ a b c+ a a + b c a + b b+ c + b b+ c c+ a + c c+ a a + b a a + b b+ c a b c+ a + b c a b+ c + a c c+ a b a a + b c b b+ c + b b c+ a c c a + b a a b+ c + b c c+ a + c a a + b a b b+ c In this the 3rd determinant is zero because the 1st and 2nd columns are identical Then a b c c a b + a b a b c b c a c + a c c b a a c b b + a c a + b a b c b c + b c c c a a a b b + c a b a b c a b c c a b + c a b a b c
11 MATRICES AND DETERMINANTS 111 Since in these 2nd, 3rd, 4th, 5th determinants are zero because of identical columns a b c c a b + ( 1)( 1) a b c c a b a b c 2 c a b Example 9: Prove that for the nth order determinant where 1 + a 1 a 2 a 3 a n a a 2 a 3 a n a 1 a a 3 a n a 1 a 2 a a n 1 + a 1 + a 2 + +a n Solution: Performing R 2 R 2 R 1,R 3 R 3 R 1,,R n R n R 1 (ie subtracting the first row from the remaining (n 1) rows) we get 1 + a 1 a 2 a 3 a 4 a n Adding C 2,C 3,,C n columns to the first column C 1 ie C 1 C 1 + C 2 + C 3 + +C n,wehave 1 + a 1 + a 2 + +a n a 2 a 3 a 4 a n Since this is an upper triangular matrix, product of the diagonal elements (1 + a 1 + a 2 + +a n ) Example 10: Solve the following equation a x c b A c b x a b a c x 0 or for what value of x, is zero (ie, matrix A is singular) Solution: Adding C 2,C 3 columns to the first column C 1, ie, C 1 C 1 + C 2 + C 3 we have a x + c + b c b c + b x + a b x a b + a + c x a c x 1 c b (a + b + c x) 1 b x a 1 a c x performing R 2 R 2 R 1,R 3 R 3 R 1 we get 1 c b (a + b + c x) 0 b x c a b 0 a c c x b (a + b + c x) 1 [(b x c)(c x b) (a c)(a b)] (a + b + c x) (x 2 a 2 b 2 c ab + bc + ca) Thus 0 when x a + b + c or x ± a 2 + b 2 + c 2 ab bc ca Example 11: Compute the product directly (a) Row by column (b) Row by row (c) Column by column (d) Column by row (e) By individually, calculating the determinants where A , B Solution: Product of the determinants (a) Row by column AB (b) Row by row AB (c) Column by column AB
12 112 MATHEMATICAL METHODS (d) Column by row AB (e) A 3, B 13, so AB (3)( 13) 39 Example 12: Find the derivative of the determinant wrt x (a) using formula (b) by differentiating the value of the (expanded) determinant wrt x d x 1 2 dx x 2 2x + 1 x 3 0 3x 2 x Solution: (a) By formula, the derivative sum of 3 determinants where the 1st, 2nd, 3rd rows are differentiated respectively wrt x Thus d x 1 2 dx x 2 2x + 1 x 3 0 3x 2 x x 2 2x + 1 x 3 0 3x 2 x x x 2 3x 2 0 3x 2 x x 1 2 x 2 2x + 1 x x [(2x + 1)(x 2 + 1) x 3 (3x 2)] + [x 2 (x 2 + 1) 3x 2 (3x 2) 1(2x(x 2 + 1) 0)+ + 2(2x(3x 2))] + [x((2x + 1)2x 3x 3 ) x 2 (2x 6)] 1 6x + 21x x 3 15x 4 (b) Expanding the given determinant x[(2x + 1)(x 2 + 1) x 3 (3x 2)] 1[x 2 (x 2 + 1) 0] + 2[x 2 (3x 2) 0] 3x 5 + 3x 4 + 7x 3 3x 2 + x So d d of determinant dx dx ( 3x5 + 3x 4 + 7x 3 3x 2 + x) 15x x x 2 6x + 1 Example 13: Determine the values of x for which matrix A is non singular given 3 x 2 2 A 2 4 x x Solution: 3 x 2 2 A det A 2 4 x x Expanding the determinant A (3 x)[(4 x)( 1 x) + 4] 2[2( 1 x) + 2) + 2[ 8 + 2(4 x)] x(x 2 6x 9) x(x 3) 2 Then A 0 when x 0 or 3 Thus matrix A is non singular ie A 0 for any x other than zero and 3 EXERCISE 1 Evaluate the following determinants (a) (b) cos θ sin θ sin θ cos θ (c) (d) 1 p p 2 1 q q 2 1 r r (e) (f) Ans (a)16(b)1(c) 126 (d) qr(r q) + rp(p r) + pq(q p) (e)4 (f) 1 2 Evaluate the determinants using triangularization (a) (b) (c) (d) Ans (a) 118 (b) 304 (c) 0 (d) 9 3 Determine the values of x for which the determinant is zero x + 2 2x + 3 3x + 4 (a) 2x + 3 3x + 4 4x + 5 3x + 5 5x x + 17
13 1 + x (b) 1 2+ x x x x + 1 2x + 1 3x + 1 (c) 2x 4x + 3 6x + 3 4x + 1 6x + 4 8x + 4 Ans (a) x 1, 1, 2 (b)x 0, 10 (c) x 0, Find the value of the determinant 1 a b+ c a b c d (a) 1 b c+ a 1 c a + b (b) a b c d a b c d a b c d 1 a a 2 1 a a 2 a 3 (c) 1 b b 2 (d) 1 b b 2 b 3 1 c c 2 1 c c 2 c 3 1 d d 2 d 3 1 aa 2 a 3 1 x 1 x1 2 x n 1 1 (e) 1 bb 2 b 3 1 x 2 x2 2 x n c c 2 c 3 (f) 1 dd 2 d 3 1 x n 1 xn 1 2 xn 1 n 1 1 x n xn 2 xn n 1 Ans (a) 0 Hint: C 3 C 3 + C 2,take(a + b + c) common) (b) 8 abcd (Hint: R 2 + R 1,R 3 + R 1,R 4 + R 1, diagonal, product of diagonal elements) (c) (a b)(b c)(c a) (Hint: a b, a c, b c, 0 Assume L(a b)(b c)(c a), determine constant L 1 by comparing the diagonal element) (d) (a b)(a c)(a d)(b c)(b d)(c d) (e) (a b)(a c)(a d)(b c)(b d) (c d)(a + b + c + d) Hint: By Factor Theorem L(a b)(a c)(a d)(b c)(b d)(c d) since at a b, a c, a d,b c, b d,c d, the determinant is zero Since principal diagonal is bc 2 d 4 is of 7th degree, introduce a linear factor (a + b + c + d), then determine L 1 MATRICES AND DETERMINANTS 113 (f) ( 1) n(n 1) 2 π where π product of factors (x i x j ) with i<j( n) Hint: At x 1 x 2,x 3,,x n,the 0so (x 1 x 2 )(x 1 x 3 ) (x 1 x n ) are factors of Similarly at x 2 x 3,x 4, x n,the 0 so (x 2 x 3 )(x 2 x 4 ) (x 2 x n ) are factors of and so on At x n 1 x n,the 0,so (x n 1 x n ) is a factor of Compare the principal diagonal term to get the value of the constant coefficient 1 + a Prove that 1 1+ b c d abcd ( a + 1 b + 1 c + 1 ) d Hint: Take a, b, c, d common from R 1,R 2,R 3,R 4 Add R 2,R 3,R 4 to R 1, take common a + 1 b + 1 c + 1 d, subtract C 1 from C 2,C 3,C 4 a 2 + x ab ac ad 6 Evaluate ab b 2 + x bc bd ac bc c 2 + x cd ad bd cd d 2 + x Ans x 3 (a 2 + b 2 + c 2 + d 2 + x) Hint: Divide by abcd, multiply C 1,C 2,C 3,C 4 by a, b, c, d Take a, b, c, d common from R 1,R 2,R 3,R 4 Add C 2,C 3,C 4 to C 1, subtract R 1 from R 2,R 3,R 4 (b + c) 2 a 2 a 2 7 Find b 2 (c + a) 2 b 2 c 2 c 2 (a + b) 2 Ans 2abc(a + b + c) 3 Hint: a,b,c, are factors (ie 0 when a 0) (a + b + c) 2 is factor since 3 columns are equal, when a + b + c 0 Principal diagonal 6thdegreeso(a + b + c) isalsoafactor a + b c c 8 Show that a b+ c a b b c+ a 4abc Hint: Expand into 8 determinants
14 114 MATHEMATICAL METHODS 9 Show that a b a b b a b a c d c d 4(a 2 + b 2 )(c 2 + d 2 ) d c d c Hint: Add C 1 to C 3,C 2 to C 4 Take2 2 4 common Then subtract C 3 from C 1,C 4 from C 2 expand 2b 1 + c 1 c 1 + 3a 1 2a 1 + 3b 1 10 Show that 2b 2 + c 2 c 2 + 3a 2 2a 2 + 3b 2 2b 3 + c 3 c 3 + 3a 3 2a 3 + 3b 3 a 1 b 1 c 1 31 a 2 b 2 c 2 a 3 b 3 c 3 Hint: Expand into 8 determinants, 27D + 4D, remaining six determinants are zeros 11 Find the product of the determinants and Ans ( 13)( 13) If A , B Then compute 15A 2 2AB B 2 without calculating A and B independently Ans Hint: A Similarly AB , B Find the derivatives of x2 x 3 2x 3x + 1 Ans 2x + 9x 2 8x 3 x 2 x Find the derivative of 1 2x 1 x 3 0 x 2 Ans 5 + 4x 12x 2 6x 5 b 2 + ac bc c 2 15 Evaluate ab 2ac bc a 2 ab b 2 + ac Ans 4a 2 b 2 c 2 b c 0 2 Hint: A a 0 c 0 a b 16 Obtain all solutions of the following equations 1 3 x (a) 2x 3 1 x 3x + 1 9x 28 2 x 2 1 x x + 2 x x x 2 x (b) 0 0 x + 2 x x 2 x Ans (a) x 1, ±3i (b) x 0, ±2
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 informationMATRIX 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 informationFundamentals 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 informationIntroduction 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 informationChapter 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[ 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 informationMATH2210 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 informationChapter 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 informationMATRICES 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 informationPrepared 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 informationMATRICES 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 informationA 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 informationMatrix & 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 informationMatrix 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 informationCLASS 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 informationCS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages:
CS100: DISCRETE STRUCTURES Lecture 3 Matrices Ch 3 Pages: 246262 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 informationEquality: 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 mbyn 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 informationElementary 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 informationA 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 information2 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 informationA 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 informationECON 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 informationELEMENTARY 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 informationIntroduction. 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 informationReview 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 informationLecture Notes in Linear Algebra
Lecture Notes in Linear Algebra Dr. Abdullah AlAzemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................
More informationMatrices. 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 informationMathematics. EC / EE / IN / ME / CE. for
Mathematics for EC / EE / IN / ME / CE By www.thegateacademy.com Syllabus Syllabus for Mathematics Linear Algebra: Matrix Algebra, Systems of Linear Equations, Eigenvalues and Eigenvectors. Probability
More informationMath Bootcamp An pdimensional 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 pdimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the
More informationMATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Final Revision CLASS XII CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION.
MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Final Revision CLASS XII 2016 17 CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.),
More informationMatrices A brief introduction
Matrices A brief introduction Basilio Bona DAUIN Politecnico di Torino Semester 1, 201415 B. Bona (DAUIN) Matrices Semester 1, 201415 1 / 41 Definitions Definition A matrix is a set of N real or complex
More informationChapter 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 information1 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 informationLinear 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 informationMatrices 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 information1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )
Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems PerOlof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical
More informationMatrices and Linear Algebra
Contents Quantitative methods for Economics and Business University of Ferrara Academic year 20172018 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 informationn 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 informationMatrix 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 informationSAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra
1.1. Introduction SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear algebra is a specific branch of mathematics dealing with the study of vectors, vector spaces with functions that
More informationa 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 informationLinear 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 informationElementary 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 informationMatrices. In this chapter: matrices, determinants. inverse matrix
Matrices In this chapter: matrices, determinants inverse matrix 1 1.1 Matrices A matrix is a retangular array of numbers. Rows: horizontal lines. A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 41 a
More information1 Determinants. 1.1 Determinant
1 Determinants [SB], Chapter 9, p.188196. [SB], Chapter 26, p.719739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to
More informationPOLI270  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 informationVectors 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 mbyn array of numbers A = a 11 a 12 a 13 a 1n a 21 a 22 a 23 a
More informationINSTITIÚID 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 2dimensional and 3dimensional graphics programming
More informationIntroduction to Matrix Algebra
Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A pdimensional 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 informationFoundations of Matrix Analysis
1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the
More informationReview from Bootcamp: Linear Algebra
Review from Bootcamp: Linear Algebra D. Alex Hughes October 27, 2014 1 Properties of Estimators 2 Linear Algebra Addition and Subtraction Transpose Multiplication Cross Product Trace 3 Special Matrices
More information10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )
c Dr. Igor Zelenko, Fall 2017 1 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections 7.27.4) 1. When each of the functions F 1, F 2,..., F n in righthand side
More informationII. Determinant Functions
Supplemental Materials for EE203001 Students II Determinant Functions ChungChin Lu Department of Electrical Engineering National Tsing Hua University May 22, 2003 1 Three Axioms for a Determinant Function
More informationMatrices and Determinants for Undergraduates. By Dr. Anju Gupta. Ms. Reena Yadav
Matrices and Determinants for Undergraduates By Dr. Anju Gupta Director, NCWEB, University of Delhi Ms. Reena Yadav Assistant Professor, NCWEB, University of Delhi Matrices A rectangular arrangement consisting
More informationLinear 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 informationLinear 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 informationMatrices A brief introduction
Matrices A brief introduction Basilio Bona DAUIN Politecnico di Torino Semester 1, 201617 B. Bona (DAUIN) Matrices Semester 1, 201617 1 / 41 Definitions Definition A matrix is a set of N real or complex
More informationSection 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 informationSystems 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 informationTOPIC III LINEAR ALGEBRA
[1] Linear Equations TOPIC III LINEAR ALGEBRA (1) Case of Two Endogenous Variables 1) Linear vs. Nonlinear Equations Linear equation: ax + by = c, where a, b and c are constants. 2 Nonlinear equation:
More informationSection 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 informationWe 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 informationFinite Mathematics Chapter 2. where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero.
Finite Mathematics Chapter 2 Section 2.1 Systems of Linear Equations: An Introduction Systems of Equations Recall that a system of two linear equations in two variables may be written in the general form
More informationSystems 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 informationSection 1.6. M N = [a ij b ij ], (1.6.2)
The Calculus of Functions of Several Variables Section 16 Operations with Matrices In the previous section we saw the important connection between linear functions and matrices In this section we will
More informationReview of Vectors and Matrices
A P P E N D I X D Review of Vectors and Matrices D. VECTORS D.. Definition of a Vector Let p, p, Á, p n be any n real numbers and P an ordered set of these real numbers that is, P = p, p, Á, p n Then P
More informationNumerical 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 informationLecture 7: Vectors and Matrices II Introduction to Matrices (See Sections, 3.3, 3.6, 3.7 and 3.9 in Boas)
Lecture 7: Vectors and Matrices II Introduction to Matrices (See Sections 3.3 3.6 3.7 and 3.9 in Boas) Here we will continue our discussion of vectors and their transformations. In Lecture 6 we gained
More informationa11 a A = : a 21 a 22
Matrices The study of linear systems is facilitated by introducing matrices. Matrix theory provides a convenient language and notation to express many of the ideas concisely, and complicated formulas are
More informationMatrices: 2.1 Operations with Matrices
Goals In this chapter and section we study matrix operations: Define matrix addition Define multiplication of matrix by a scalar, to be called scalar multiplication. Define multiplication of two matrices,
More informationLinear Equations and Matrix
1/60 ChiaPing Chen Professor Department of Computer Science and Engineering National Sun Yatsen University Linear Algebra Gaussian Elimination 2/60 Alpha Go Linear algebra begins with a system of linear
More information7.3. Determinants. Introduction. Prerequisites. Learning Outcomes
Determinants 7.3 Introduction Among other uses, determinants allow us to determine whether a system of linear equations has a unique solution or not. The evaluation of a determinant is a key skill in engineering
More informationGraduate 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 informationLinear 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 informationMath 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 informationIntroduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX
Introduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX September 2007 MSc Sep Intro QT 1 Who are these course for? The September
More informationMAT 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 informationMaterials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat
Materials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat Linear Algebra Lecture 2 1.3.7 Matrix Matrix multiplication using Falk s
More informationMATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.
MATH 311504 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 informationAlgebra & Trig. I. For example, the system. x y 2 z. may be represented by the augmented matrix
Algebra & Trig. I 8.1 Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural
More informationMatrix 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 informationChapter 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 informationMatrices. Math 240 Calculus III. Wednesday, July 10, Summer 2013, Session II. Matrices. Math 240. Definitions and Notation.
function Matrices Calculus III Summer 2013, Session II Wednesday, July 10, 2013 Agenda function 1. 2. function function Definition An m n matrix is a rectangular array of numbers arranged in m horizontal
More informationA Review of Matrix Analysis
Matrix Notation Part Matrix Operations Matrices are simply rectangular arrays of quantities Each quantity in the array is called an element of the matrix and an element can be either a numerical value
More informationOR MSc Maths Revision Course
OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:3012:30 Mathematics revision
More informationCHAPTER 6. Direct Methods for Solving Linear Systems
CHAPTER 6 Direct Methods for Solving Linear Systems. Introduction A direct method for approximating the solution of a system of n linear equations in n unknowns is one that gives the exact solution to
More informationMATH 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 informationMatrix 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 informationLinear Algebra and Matrices
Linear Algebra and Matrices 4 Overview In this chapter we studying true matrix operations, not element operations as was done in earlier chapters. Working with MAT LAB functions should now be fairly routine.
More informationMATH 106 LINEAR ALGEBRA LECTURE NOTES
MATH 6 LINEAR ALGEBRA LECTURE NOTES FALL  These Lecture Notes are not in a final form being still subject of improvement Contents Systems of linear equations and matrices 5 Introduction to systems of
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationMATRICES AND ITS APPLICATIONS
MATRICES AND ITS Elementary transformations and elementary matrices Inverse using elementary transformations Rank of a matrix Normal form of a matrix Linear dependence and independence of vectors APPLICATIONS
More informationELEMENTARY 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 informationUndergraduate 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 informationChapter 1 Matrices and Systems of Equations
Chapter 1 Matrices and Systems of Equations System of Linear Equations 1. A linear equation in n unknowns is an equation of the form n i=1 a i x i = b where a 1,..., a n, b R and x 1,..., x n are variables.
More informationIntroduction to Matrices and Linear Systems Ch. 3
Introduction to Matrices and Linear Systems Ch. 3 Doreen De Leon Department of Mathematics, California State University, Fresno June, 5 Basic Matrix Concepts and Operations Section 3.4. Basic Matrix Concepts
More informationSAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra
SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to 1.1. Introduction Linear algebra is a specific branch of mathematics dealing with the study of vectors, vector spaces with functions that
More information