8.6. THE INVERSE OF A MATRIX 149

Example 8.6.8 Let A =

 1 2 21 0 22 2 4

. Find A−1 if it exists.

Write the augmented matrix (A|I) 1 2 2 | 1 0 01 0 2 | 0 1 02 2 4 | 0 0 1

and proceed to do row operations attempting to obtain

(I|A−1

). Take (−1) times the top

row and add to the second. Then take (−2) times the top row and add to the bottom. 1 2 2 | 1 0 00 −2 0 | −1 1 00 −2 0 | −2 0 1

Next add (−1) times the second row to the bottom row. 1 2 2 | 1 0 0

0 −2 0 | −1 1 00 0 0 | −1 −1 1

At this point, you can see there will be no inverse because you have obtained a row of zerosin the left half of the augmented matrix (A|I) . Thus there will be no way to obtain I on theleft.

Example 8.6.9 Let A =

 1 0 11 −1 11 1 −1

. Find A−1 if it exists.

▶▶Form the augmented matrix 1 0 1 | 1 0 0

1 −1 1 | 0 1 01 1 −1 | 0 0 1

 .

Now do row operations until the n×n matrix on the left becomes the identity matrix. Thisyields after some computations,

1 0 0 | 0 12

12

0 1 0 | 1 −1 00 0 1 | 1 − 1

2 − 12

