955 9.7 Properties of Matrices Now use row 1 of A and column 2 of B to determine the element in row 1, column 2 of the product matrix. c -3 5 4 0 2 4d £ -6 2 3 4 3 -2§ -3142 + 4132 + 21-22 = -4 Next, use row 2 of A and column 1 of B. This will give the row 2, column 1 element of the product matrix. c -3 5 4 0 2 4d £ -6 2 3 4 3 -2§ 51-62 + 0122 + 4132 = -18 Finally, use row 2 of A and column 2 of B to find the element for row 2, column 2 of the product matrix. c -3 5 4 0 2 4d £ -6 2 3 4 3 -2§ 5142 + 0132 + 41-22 = 12 The product matrix can be written using the four elements just found. c -3 5 4 0 2 4d £ -6 2 3 4 3 -2§ = c 32 -18 -4 12d Product AB NOTE As seen here, the product of a 2 * 3 matrix and a 3 * 2 matrix is a 2 * 2 matrix. The dimension of a product matrix AB is given by the number of rows of A and the number of columns of B, respectively. By definition, the product AB of an m* n matrix A and an n * p matrix B is found as follows. To find the ith row, jth column element of AB, multiply each element in the ith row of A by the corresponding element in the jth column of B. (Note the shaded areas in the matrices below.) The sum of these products will give the row i, column j element of AB. A = F a11 a12 a13 g a1n a21 a22 a23 g a2n f ai1 ai2 ai3 g ain f am1 am2 am3 g amnV B = D b11 b12 g b1j g b1p b21 b22 g b2j g b2p f bn1 bn2 g bnj g bnpT Matrix Multiplication The number of columns of an m* n matrix A is the same as the number of rows of an n * p matrix B (i.e., both n). The element cij of the product matrix C = AB is found as follows. cij =ai1b1j +ai2b2j +P+ainbnj Matrix AB will be an m* p matrix. 3 4 2 5 0 4d 5 0 4d
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