956 CHAPTER 9 Systems and Matrices EXAMPLE 5 Deciding Whether Two Matrices Can Be Multiplied Suppose A is a 3 * 2 matrix, while B is a 2 * 4 matrix. (a) Can the product AB be calculated? (b) If AB can be calculated, what is its dimension? (c) Can BA be calculated? (d) If BA can be calculated, what is its dimension? SOLUTION (a) The following diagram shows that AB can be calculated because the number of columns of A is equal to the number of rows of B. (Both are 2.) Matrix A Matrix B 3 * 2 2 * 4 Must match Dimension of AB 3 * 4 (b) As indicated in the diagram above, the product AB is a 3 * 4 matrix. (c) The diagram below shows that BA cannot be calculated. Matrix B Matrix A 2 * 4 3 * 2 Different (d) The product BA cannot be calculated because B has 4 columns and A has only 3 rows. S Now Try Exercises 49 and 51. EXAMPLE 6 Multiplying Matrices Let A = c 1 7 -3 2d and B = c 1 3 0 1 -1 4 2 -1d . Find each product, if possible. (a) AB (b) BA SOLUTION (a) First decide whether AB can be found. AB = c 1 7 -3 2d c 1 3 0 1 -1 4 2 -1d A is 2 * 2 and B is 2 * 4, so the product will be a 2 * 4 matrix. = c 1112 + 1-323 7112 + 2132 1102 + -321 7102 + 2112 11-12 + 1-324 71-12 + 2142 1122 + 1-321-12 7122 + 21-12 d Use the definition of matrix multiplication. = c -8 13 -3 2 -13 1 5 12d Perform the operations. (b) B is a 2 * 4 matrix, and A is a 2 * 2 matrix, so the number of columns of B (here 4) does not equal the number of rows of A (here 2). Therefore, the product BA cannot be calculated. S Now Try Exercises 69 and 73. The three screens here support the results of the matrix multiplication in Example 6. The final screen indicates that the product BA cannot be found.
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