This demonstrates the utility of using row operations

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Unformatted text preview: efined, the number of entries in the rows of A must match the number of entries in the columns of B . This means that the number of columns of A must match the number of rows of B .7 In other words, to multiply A times B , the second dimension of A must match the first dimension of B , which is why in Definition 8.10, Am×n is being multiplied by a matrix Bn×r . Furthermore, the product matrix AB has as many rows as A and as many columns of B . As a result, when multiplying a matrix Am×n by a matrix Bn×r , the result is the matrix ABm×r . Returning to our example matrices below, we see that A is a 2 × 3 matrix and B is a 3 × 4 matrix. This means that the product matrix AB is defined and will be a 2 × 4 matrix. 3 1 2 −8 2 0 −1 8 −5 9 A= B= 4 −10 3 5 5 0 −2 −12 7 The reader is encouraged to think this through carefully. 8.3 Matrix Arithmetic 483 Using Ri to denote the ith row of A and Cj to denote the j th column of B , we form AB according to Definition 8.10. AB = R1 · C 1 R1 · C 2 R1 · C 3 R1 · C 4 R2 · C 1 R2 · C 2 R2 · C...
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