969 9.8 Matrix Inverses Step 2 There is already a 1 in the upper left-hand corner as desired. Begin by using the row transformation that will result in 0 for the first element in the second row. Multiply the elements of the first row by -2, and add the result to the second row. £1 0 3 0 -2 0 1 -3 0 3 1 -2 0 0 1 0 0 0 1§ -2R1 + R2 We introduce 0 for the first element in the third row by multiplying the elements of the first row by -3 and adding to the third row. £1 0 0 0 -2 0 1 -3 -3 3 1 -2 -3 0 1 0 0 0 1§ -3R1 + R3 To obtain 1 for the second element in the second row, multiply the elements of the second row by - 1 2 . £1 0 0 0 1 0 1 3 2 -3 3 1 1 -3 0 - 1 2 0 0 0 1§ - 1 2 R2 We want 1 for the third element in the third row, so multiply the elements of the third row by - 1 3 . C 1 0 0 0 1 0 1 3 2 1 3 1 1 1 0 - 1 2 0 0 0 - 1 3S - 1 3 R3 The third element in the first row should be 0, so multiply the elements of the third row by -1 and add to the first row. ≥1 0 0 0 1 0 0 3 2 1 4 0 1 1 0 - 1 2 0 1 3 0 - 1 3¥ -1R3 + R1 Finally, to introduce 0 as the third element in the second row, multiply the elements of the third row by - 3 2 and add to the second row. D1 0 0 0 1 0 0 0 1 4 0 - 1 2 1 0 - 1 2 0 1 3 1 2 - 1 3T - 3 2 R3 + R2 Step 3 The last transformation shows the inverse. A-1 = D 0 - 1 2 1 0 - 1 2 0 1 3 1 2 - 1 3T Confirm this by forming the product A-1A or AA-1, each of which should equal the matrix I3. S Now Try Exercises 17 and 25. A graphing calculator can be used to find the inverse of a matrix. This screen supports the result in Example 2. The elements of the inverse are expressed as fractions, so it is easier to compare with the inverse matrix found in the example. If the inverse of a matrix exists, it is unique. That is, any given square matrix has no more than one inverse. The proof of this is left as Exercise 69.
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