![2 3
111
A 1 1 and B
1 0 2
1 0
[3 x 2] [2 x 3]
A and B can be multiplied
[3 x 3]](https://image.slidesharecdn.com/week2-100401102242-phpapp02/85/Week2-29-320.jpg)
Week2
- 3. A matrix is an ordered rectangular array of numbers. The size of a matrix is given by the number of rows and the number of columns. Let m denote the number of rows Let n denote the number of columns. Let aij denote the entry in the ith row and the jth column.
- 4. m n a11 a12 a1n a21 a22 a2 n m rows am1 am 2 amn n columns
- 5. Zero matrix ij square matrix
- 7. The Identity Matrix, denoted I n , is a diagonal matrix of order nxn with all the diagonal entries equal to 1.
- 8. 0 0 0 d11 0 0 0 0 0 0 d 22 0 O D 0 0 0 0 0 d mm 1 0 0 0 1 0 I Let I n denote an n n identity matrix 0 0 1
- 9. A square matrix all its elements below the main diagonal are zeros. 0 i j aij 0 i j 2 3 3 A 0 7 1 0 0 9
- 10. A square matrix all its elements above the main diagonal are zeros. 0 i j aij 0 i j 2 0 0 A 5 7 0 2 8 9
- 11. The matrix A is 3x3 of the form 2i i>j aij i j i j a11 a12 a13 2 3 4 A 4 4 5 A a21 a22 a23 6 6 6 a31 a32 a33
- 12. 2i i j aij 5 i j i j i j 3j i j bij i j i j If A and B are 3x3 find A-2B
- 13. a11 a12 a13 5 1 2 A a 21 a 22 a 23 4 5 1 a 31 a 32 a 33 6 6 5 3 3 4 B 3 6 5 3 6 9 1 7 10 A 2B 2 7 11 0 6 13
- 14. Equality of Matrices: Two matrices are equal if they have the same size and their corresponding entries are equal.
- 15. Find x and y that satisfies the following equation x y y y y y=2 x+6=2 x=-4
- 16. A new matrix C may be defined as the additive combination of matrices A and B where: C = A + B is defined by: ij ij ij i 1, 2,..., m and j 1, 2,..., n Note: Only matrices of the same dimension can be added
- 17. Addition of Matrices a11 a12 a1n b11 b12 b1n a21 a22 a2 n b21 b22 b2 n A and B am1 am 2 amn bm1 bm 2 bmn a11 b11 a12 b12 a1n b1n a21 b21 a22 b22 a2 n b2 n A B am1 bm1 am 2 bm 2 amn bmn
- 18. 3 4 1 2 4 6 A B C 5 6 3 4 8 10
- 19. Multiplying a matrix by a real number (scalar) results in a matrix with each entry multiplied by the scalar. Let k be a real number. a11 a12 a1n ka11 ka12 ka1n a21 a22 a2 n ka21 ka22 ka2 n A kA am1 am 2 amn kam1 kam 2 kamn
- 21. k(A + B) = kA + k B (x+y) A = x A + y A (x y) A = x ( y A) oA=O xO=O
- 22. C = A - B is defined by C = A + (-1) B a11 b11 a12 b12 a1n b1n a21 b21 a22 b22 a2n b2n A B am1 bm1 am2 bm2 amn bmn
- 23. The scalar matrix can be written as . C= In x 0 0 0 0 x 0 0 0 0 x 0 0 0 0 x
- 24. Matrices A and B have these dimensions:
- 25. Matrices A and B can be multiplied if:
- 26. The resulting matrix will have the dimensions:
- 27. a11 a12 a1n b11 b12 b1 p a21 a22 a2 n b21 b22 b2 p A and B am1 am 2 amn bn1 bn 2 bnp c11 c12 c1 p Entry cij is obtained by c21 c22 c2 p taking the sum of the AB C products of the entries of cm1 cm 2 cmp the ith row in A with the jth column in B. Entry c11 a11b11 a12b21 ... a1nbn1 c21 a21b11 a22b21 ... a2 nbn1 c22 a11b12 a12b22 ... a1nbn 2
- 28. mxn mxs sxn s c ij aik b kj k 1 i 1, 2,..., m j 1, 2,..., n
- 29. 2 3 111 A 1 1 and B 1 0 2 1 0 [3 x 2] [2 x 3] A and B can be multiplied [3 x 3]
- 30. 2*1 3*1 5 2*1 3*0 2 2*1 3*2 8 5 2 8 C 1*1 1*1 2 1*1 1*0 1 1*1 1*2 3 2 1 3 1*1 0*1 1 1*1 0*0 1 1*1 0*2 1 1 1 1
- 31. 2x3 3x2 2x2 2 3 1 1 1 BxA 1 1 1 0 2 1 0 4 4 = 4 3
- 32. Properties of Matrix Multiplication 1.AB BA 2- AB C A BC 3. A (B C ) AB AC (B+C)A=BA+CA 4. If A is an m n matrix and I m and I n are m m and n n identity matrices, respectively, then I m A AI n A
- 33. 5- Amxn 0nxk =0mxk 0kxm Amxn = 0kxn
- 35. If A is an m x n matrix with elements aij, then the transpose of A, denoted, AT, is an n x m matrix with elements aji. a11 a12 a1n a11 a21 am1 a21 a22 a2 n T a12 a22 am 2 A A am1 am 2 amn a1n a2 n amn
- 36. T T T T T T T T T T
- 37. A square matrix B is said to be symmetric if B = BT brs = bsr s r r=1,2, ,m s=1,2, ,n
- 38. Any diagonal matrix is symmetric A squared zero matrix is symmetric The transpose of an upper triangular matrix is a lower triangular matrix
- 39. A squared matrix is said to skew-symmetric if A = - AT
- 40. 0 1 2 A 1 0 3 2 3 0 0 1 2 T A 1 0 3 2 3 0 T A A
- 41. The common term of a skew-symmetric matrix can be written as r s ars asr r s
- 42. If the common element of a 3x3 matrix is given by rs Show that A is a skew-symmetric matrix
- 43. 0 1 1 A 1 0 1 1 1 0 0 1 1 T A 1 0 1 1 1 0 T A A
- 44. X =(x1,x2, .,xn) 1xn AT =(1,1, .,1) 1xn xi = X A = AT XT xi2 = X XT
- 45. A square matrix is said to be orthogonal if T T T H E I D E N T I T Y M AT R I X I S O R T H O G O N AL AN D S YM M E T R I C
- 46. -1 -1 2 2 A = 1 -1 2 2 1 -1 -1 A = 2 1 -1 T 1 -1 1 A = 2 -1 -1
- 47. T T 2
- 48. If A is a square matrix and k is a positive integer, the power k of A is defined as Ak = A . A ..A So, A2 = A . A and A0 = I
- 49. A square matrix A is said to be idempotent if Ak = A for any positive integer k The identity matrix is idempotent The square zero matrix is idempotent
- 50. Show that the matrix A defined below is idempotent 1 1 2 2 A 1 1 2 2 1 1 1 1 1 1 2 2 2 2 2 2 2 A AA A 1 1 1 1 1 1 2 2 2 2 2 2 3 A A 2A AA A
- 51. Assume Ak-1 = A Ak = Ak-1 A =A A=A So Ak = A for any positive integer k i.e. A is idempotent
- 52. Definition: Let A be an n n matrix. An inverse of A is an n n matrix B such that: AB = In and BA = In A is then called invertible. B is called an inverse of A If A has no inverse, it is called singular. .
- 53. The inverse (if exists) is unique. Proof: Assume A is invertible, with two inverses B and C, i.e. AB=BA=I and AC = CA = I, (BA)C=B(AC)=BI=B (BA)C=IC=C Thus, B=C
- 54. Since the inverse ,if it exists, is unique, we call it A-1 A A-1 = A-1 A = In
- 55. Let A and B be invertible matrices of the same size, and k be a nonzero scalar. Then: 1. I 1 =I 2. (A 1) 1 = A 3. (kA) 1 = k 1A 1
- 56. 4- (AB) 1 = B 1A 1 5- (An) 1 = (A 1)n 6- (AT) 1 = (A 1)T 7- If A is orthogonal then it is invertible and A-1=AT
- 57. If A and B are nxn invertible matrices, prove that AB is invertible also. (AB)(AB)-1=(AB)(B-1A-1) = A(BB-1)A-1 -1 -1 n n (AB)-1 (AB) = (B-1A-1) (AB)= B(AA-1)B-1 -1 -1 n n
- 58. Find the inverse of the 2 2 matrix:
- 59. 1 1
- 60. 1
- 61. 3 1 If A 5 2 Show that A2 -5A+I2=0 Using this result find A-1
- 62. A2 - 5A + I2 =0 A-1 A A 5 A-1 A + A-1 = 0 A 5 I + A-1 = 0 A-1 =5I A -1
- 63. 3 5 If A = 2 3 1 4 7 B 2 4 Fin d a m a trix X su ch th a t XB =A
- 64. XB=A X B B-1 = A B-1 X I = A B-1
- 65. Let A, B, and X be 3 invertible matrices of the same size. Solve the following matrix equation for X: (A 1XB) 1 = (BA)2 Note: Be careful with the order of the matrix multiplication. Answer: X = (B2AB) 1 = B 1A 1(B 1)2
- 66. (A-1 X B)-1 = (B A)2 B-1 X-1 (A-1 )-1 = (B A)2 B-1 X-1 A = (BA)2 B B-1 X-1 A = B (BA)2 I X-1 A A-1 = B (BA)2 A-1 X-1 I = B BA BA A-1 X-1 = B2 A B X = ( B2 A B)-1
- 67. Show that the inverse of the general 2 2 matrix: a b A c d 1 d b A ad bc c a 1 1
- 68. -1 x y A s s u m e A = w z -1 a b x y 1 0 A A = = c d w z 0 1 a x + b w = 1 c x + d w = 0 -c a x - c b w = -c c a x + a d w = 0 -c w = a d - b c 0 a d - b c d x = a d -b c 0 ad b c