968 CHAPTER 9 Systems and Matrices CHECK Multiply A by A-1. The result should be I 2. AA-1 = c 2 1 4 -1d c 1 6 1 6 2 3 - 1 3d = c 1 3 + 2 3 1 6 - 1 6 4 3 - 4 3 2 3 + 1 3d = c 1 0 0 1d = I2 ✓ Thus, A-1 = c 1 6 1 6 2 3 - 1 3d . This process is summarized below. Finding an Inverse Matrix To obtain A-1 for any n * n matrix A for which A-1 exists, follow these steps. Step 1 Form the augmented matrix 3A 0 In4, where In is the n * n identity matrix. Step 2 Perform row transformations on 3A 0 In4 to obtain a matrix of the form 3In 0 B4. Step 3 Matrix B is A-1. As illustrated by the examples, the most efficient order for the transformations in Step 2 is to make the changes column by column from left to right, so that for each column the required 1 is the result of the first change. Next, perform the steps that obtain 0s in that column. Then proceed to the next column. NOTE To confirm that two n * n matrices A and B are inverses of each other, it is sufficient to show that AB = In. It is not necessary to show also that BA = In. EXAMPLE 2 Finding the Inverse of a 3 : 3 Matrix Find A-1 if A = £ 1 2 3 0 -2 0 1 -1 0§. SOLUTION Use row transformations as follows. Step 1 Write the augmented matrix 3A0 I34. £1 2 3 0 -2 0 1 -1 0 3 1 0 0 0 1 0 0 0 1§
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