957 9.7 Properties of Matrices EXAMPLE 7 Multiplying Square Matrices in Different Orders Let A = c 1 -2 3 5d and B = c -2 0 7 2d . Find each product. (a) AB (b) BA SOLUTION (a) AB = c 1 -2 3 5d c -2 0 7 2d = c 11-22 + 3102 -21-22 + 5102 1172 + 3122 -2172 + 5122 d Multiply elements of each row of A by elements of each column of B. = c -2 4 13 -4d (b) BA = c -2 0 7 2d c 1 -2 3 5d = c -2112 + 71-22 0112 + 21-22 -2132 + 7152 0132 + 2152 d Multiply elements of each row of B by elements of each column of A. = c -16 -4 29 10d Note that AB≠BA. S Now Try Exercise 79. When multiplying matrices, it is important to pay special attention to the dimensions of the matrices as well as the order in which they are to be multiplied. Examples 5 and 6 showed that the order in which two matrices are to be multiplied may determine whether their product can be found. Example 7 showed that even when both products AB and BA can be found, they may not be equal. Noncommutativity of Matrix Multiplication In general, if A and B are matrices, then AB 3BA. Matrix multiplication is not commutative. Properties of Matrix Multiplication If A, B, and C are matrices such that all the following products and sums exist, then these properties hold true. 1 AB2C =A1BC2, A1B +C2 =AB +AC, 1B +C2A =BA +CA Matrix multiplication does satisfy the associative and distributive properties. For proofs of the first two results for the special cases when A, B, and C are square matrices, see Exercises 93 and 94. The identity and inverse properties for matrix multiplication are discussed later.
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