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DEFINITION: If A and B are m × n matrices, then the sum A+B is the m×n matrix whose entries are the sums of the corresponding entries of A and B. EXAMPLE: 1 −2 −1 0 1 2 1 −1 1 + = −2 3 −2 −2 −4 −1 −4 −1 −3 REMARK: We can add matrices only of the same size. EXAMPLE: 1 2 1 2 + 1 0 =??? 3 4 3 4 DEFINITION: If r is a scalar and A is a matrix, then the scalar multiple rA is the matrix whose entries are r times the corresponding entries in A. EXAMPLE: 1 2 −3 −2 −4 6 (−2) = −1 0 −2 2 0 4 PROPERTIES: Let A, B, and C be matrices of the same size, and let r and s be scalars. Then (a) A + B = B + A (b) (A + B) + C = A + (B + C) (c) r(A + B) = rA + rB (d) (r + s)A = rA + sA (e) r(sA) = (rs)A PROBLEM: Consider the following matrices: 1 0 3 −2 1 2 3 . A= ,B = 2 1 ,C = 2 5 2 1 4 3 2 If possible, compute: (a) AB (b) AC + B 2 (c) AB + C 2 SOLUTION: We have: (a) AB = 1 0 14 8 1 2 3 2 1 = . 16 9 2 1 4 3 2 (b) Impossible. (c) AB + C 2 3 −2 3 −2 14 8 + = 2 5 2 5 16 9 = = 5 −16 14 8 + 16 21 16 9 19 −8 . 32 30 PROPERTIES: Let A be an m × n matrix, and let B and C have sizes for which the indicated sums and products are defined. Then (a) A(BC) = (AB)C (b) A(B + C) = AB + AC (c) (B + C)A = BA + CA (d) r(AB) = (rA)B = A(rB) WARNING 1. In general, AB 6= BA. EXAMPLE: 1 2 1 1 Let A = and B = . Then 0 1 2 0 AB = BA = 1 2 0 1 1 1 2 0 1 1 2 0 1 2 0 1 = So, AB 6= BA. = 5 1 2 0 1 3 2 4 WARNING 2. If AB = AC, then it is not true in general that B = C. EXAMPLE: Let 1 0 0 0 0 0 A= ,B = , and C = . 0 0 0 1 0 2 Then AB = AC = 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 2 = = 0 0 0 0 0 0 0 0 So, AB = AC, but B 6= C. WARNING 3. If AB = 0, then it is not true in general that A = 0 or B = 0. EXAMPLE: 1 0 0 0 and B = . Then Let A = 0 0 0 1 AB = 1 0 0 0 0 0 0 1 = 0 0 0 0 So, AB = 0, but A 6= 0 and B 6= 0. THE TRANSPOSE OF A MATRIX DEFINITION: Let A be an m × n matrix. The transpose of A is the n × m matrix, denoted by AT , whose columns are formed from the corresponding rows of A. EXAMPLE: 1 2 A= 3 4 −3 1 B = 4 7 8 −5 C= 1 2 3 4 5 6 7 8 AT = 1 3 2 4 −3 4 8 1 7 −5 1 5 2 6 T C = 3 7 4 8 BT = PROPERTIES: Let A and B denote matrices whose sizes are appropriate for the following sums and products. Then (a) (AT )T = A (b) (A + B)T = AT + B T (c) (rA)T = rAT for any scalar r (d) (AB)T = B T AT THE INVERSE OF A MATRIX DEFINITION: The identity matrix I is the n×n matrix of the form 1 0 0 ... 0 0 0 1 0 ... 0 0 0 0 1 ... 0 0 I= ............ 0 0 0 ... 1 0 0 0 0 ... 0 1 MAIN PROPERTY: AI = IA = A DEFINITION: An n × n matrix A is said to be invertible if there is an n × n matrix C such that CA = I and AC = I. In this case, C is an inverse of A and is denoted by A−1. So, A−1A = I and AA−1 = I. EXAMPLE: 2 5 −7 −5 −1 Let A = . Then A = . −3 −7 3 2 In fact, we have 2 5 −7 −5 1 0 −1 AA = = −3 −7 3 2 0 1 and A−1A = −7 −5 3 2 2 5 −3 −7 = 1 0 . 0 1 THEOREM: a b Let A = . If ad − bc 6= 0, then A c d is invertible and 1 d −b −1 A = . −c a ad − bc If ad − bc = 0, then A is not invertible. PROBLEM: Solve the following system of equations: ( x1 − 2x2 = 0 x1 + 4x2 = 6 SOLUTION: We have: Ax̄ = B ⇒ x̄ = A−1B, therefore x̄ = x1 x2 = −1 1 −2 0 1 4 6 1 4 2 0 = 6 6 −1 1 1 12 = 6 6 2 = . 1