967 9.8 Matrix Inverses Use matrix multiplication for this product, which is repeated in the margin. c 2 1 4 -1d c x z y wd = c 1 0 0 1d c 2x + 4z x - z 2y + 4w y - w d = c 1 0 0 1d Set the corresponding elements equal to obtain a system of equations. 2x + 4z = 1 (1) 2y + 4w = 0 (2) x - z = 0 (3) y - w = 1 (4) Because equations (1) and (3) involve only x and z, while equations (2) and (4) involve only y and w, these four equations lead to two systems of equations. 2x + 4z = 1 x - z = 0 and 2y + 4w = 0 y - w = 1 Write the two systems as augmented matrices. c 2 1 4 -1 2 1 0d and c 2 1 4 -1 2 0 1d Each of these systems can be solved by the Gauss-Jordan method. However, since the elements to the left of the vertical bar are identical, the two systems can be combined into one matrix. c 2 1 4 -1 2 1 0d and c 2 1 4 -1 2 0 1d yields c 2 1 4 -1 2 1 0 0 1d We can solve simultaneously using matrix row transformations. We need to change the numbers on the left of the vertical bar to the 2 * 2 identity matrix. c 1 2 -1 4 2 0 1 1 0d Interchange R1 and R2 to introduce 1 in the upper left-hand corner. c 1 0 -1 6 2 0 1 1 -2d -2R1 + R2 c1 0 -1 1 3 0 1 6 1 - 1 3d 1 6 R2 c 1 0 0 1 3 1 6 1 6 2 3 - 1 3d R2 + R1 The numbers in the first column to the right of the vertical bar in the final matrix give the values of x and z. The second column gives the values of y and w. That is, c 1 0 0 1 2 x z y wd = c 1 0 0 1 2 1 6 1 6 2 3 - 1 3d so that A-1 = c x z y wd = c 1 6 1 6 2 3 - 1 3d .
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