ICS 141: Discrete Mathematics I (Fall 2014) 2.6 Matrices A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m × n matrix. 2 1 2 3 3 6 4 5 6 7 6 7 is a 4 × 3 matrix. 4 7 8 9 5 10 11 12 We specify an element of the matrix with mi;j where the element is at row i and column j. For example, m2;3 = 6. Matrix Equality Matrices are equal if and only if they have the same number of rows, the same number of columns, and the same elements at every index. Matrix Addition We can only perform matrix addition if the matrices have the same dimensions. To express the addition of two matrices, A and B, we write A + B = [ai;j + bi;j]. Then simply add the values at corresponding indices. 21 2 33 29 8 73 2(1 + 9) (2 + 8) (3 + 7)3 210 10 103 44 5 65 + 46 5 45 = 4(4 + 6) (5 + 5) (6 + 4)5 = 410 10 105 7 8 9 3 2 1 (7 + 3) (8 + 2) (9 + 1) 10 10 10 Matrix Multiplication If A and B are matrices, we can write AB to denote their multiplication. Matrix multiplication is not commutative (AB 6= BA). We can only multiply matrices if and only if the first matrix has the same number of columns as the number of rows in the second matrix. AB = C A is an i × k matrix B is an k × j matrix The result C is an i × j matrix Ci;j = (ai;1b1;j) + (ai;2b2;j) + (ai;2b2;j) + ::: + (ai;kbk;j) 1 2 5 6 (1 · 5 + 2 · 7) (1 · 6 + 2 · 8) 19 22 = = 3 4 7 8 (3 · 5 + 4 · 7) (3 · 6 + 4 · 8) 43 50 1 ICS 141: Discrete Mathematics I (Fall 2014) Identity Matrix The identity matrix is a n × n square matrix where the main diagonal consist of all ones and zeros elsewhere. 21 0 ::: 03 60 1 ::: 07 6 7 In = 6. .. .7 4. .5 0 0 ::: 1 Inverse Matrix For a square matrix A, the inverse written as A−1. If A is multiplied by A−1, then the result is the −1 −1 identity matrix. AA = A A = In. Note that not all square matrices have an inverse. Power of Matrices Square matrices can be multiplied by themselves repeatedly because they have the same number of rows and columns. An n × n matrix A raised to the positive integer k is defined as Ak = AAA : : : A A0 = I Transpose of Matrices The Transpose of A, denoted by At, is obtained by turning all the rows into columns and vice versa. Zero-One Matrices Zero-one matrices are matrices that only contain 0 or 1. Join: A _ B = [aij _ bij] Meet: A ^ B = [aij ^ bij] Boolean Product: Denoted by A B, where cij = (ai1 ^ b1j) _ (ai2 ^ b2j) _ ::: _ (aik ^ bkj) 2.6 pg 184 # 3 Find AB if 2 1 0 4 a) A = ;B = 3 2 1 3 2 1 0 4 (2 · 0 + 1 · 1) (2 · 4 + 1 · 3) 1 11 = = 3 2 1 3 (3 · 0 + 2 · 1) (3 · 4 + 2 · 3) 2 18 2 ICS 141: Discrete Mathematics I (Fall 2014) 21 −13 3 2 −1 b) A = 0 1 ;B = 4 5 1 0 3 2 3 21 −13 2(1 · 3 + (−1) · 1) (1 · 2 + (−1) · 0) (1 · (−1) + (−1) · 3)3 3 2 −1 0 1 = (0 · 3 + 1 · 1) (0 · 2 + 1 · 0) (0 · (−1) + 1 · 3) 4 5 1 0 3 4 5 2 3 (2 · 3 + 3 · 1) (2 · 2 + 3 · 0) (2 · (−1) + 3 · 3) 22 2 −43 = 41 0 3 5 9 4 7 2.6 pg 185 # 15 Let 1 1 A = 0 1 Find a formula for An, whenever n is a positive integer. 1 1 1 1 1 2 A2 = AA = = 0 1 0 1 0 1 1 1 1 2 1 3 A3 = AA2 = = 0 1 0 1 0 1 1 1 1 3 1 4 A4 = AA3 = = 0 1 0 1 0 1 . 1 1 1 n − 1 1 n An = AAn−1 = = 0 1 0 1 0 1 2.6 pg 185 # 19 Let A be the 2 × 2 matrix a b A = c d Show that if ad − bc 6= 0, then 2 d −b 3 −1 6ad − bc ad − bc7 A = 4 −c a 5 ad − bc ad − bc 2 d −b 3 2ad − bc −ab + ba3 a b 1 0 AA−1 = 6ad − bc ad − bc7 = 6ad − bc ad − bc 7 = c d 4 −c a 5 4cd − cd −bc + ad5 0 1 ad − bc ad − bc ad − bc ad − bc 3 ICS 141: Discrete Mathematics I (Fall 2014) 2 d −b 3 2 ad − bc db − bd 3 a b 1 0 A−1A = 6ad − bc ad − bc7 = 6 ad − bc ad − bc 7 = 4 −c a 5 c d 4−ca + ac −bc + ad5 0 1 ad − bc ad − bc ad − bc ad − bc Extra Problem Let 21 1 1 33 A = 42 0 4 65 1 1 3 7 What is At? 21 2 13 t 61 0 17 A = 6 7 41 4 35 3 6 7 2.6 pg 185 # 27 21 0 13 20 1 13 Let A = 41 1 05 and B = 41 0 15 Find 0 0 1 1 0 1 a) A _ B 21 0 13 20 1 13 21 1 13 41 1 05 _ 41 0 15 = 41 1 15 0 0 1 1 0 1 1 0 1 b) A ^ B 21 0 13 20 1 13 20 0 13 41 1 05 ^ 41 0 15 = 41 0 05 0 0 1 1 0 1 0 0 1 c) A B 21 0 13 20 1 13 41 1 05 41 0 15 0 0 1 1 0 1 2(1 ^ 0) _ (0 ^ 1) _ (1 ^ 1) (1 ^ 1) _ (0 ^ 0) _ (1 ^ 0) (1 ^ 1) _ (0 ^ 1) _ (1 ^ 1)3 = 4(1 ^ 0) _ (1 ^ 1) _ (0 ^ 1) (1 ^ 1) _ (1 ^ 0) _ (0 ^ 0) (1 ^ 1) _ (1 ^ 1) _ (0 ^ 1)5 (0 ^ 0) _ (0 ^ 1) _ (1 ^ 1) (0 ^ 1) _ (0 ^ 0) _ (1 ^ 0) (0 ^ 1) _ (0 ^ 1) _ (1 ^ 1) 21 1 13 = 41 1 15 1 0 1 4.