148 CHAPTER 8. MATRICES

Example 8.6.7 Let A =

 1 2 21 0 23 1 −1

. Find A−1 if it exists.

Set up the augmented matrix (A|I) 1 2 2 | 1 0 01 0 2 | 0 1 03 1 −1 | 0 0 1

Next take (−1) times the first row and add to the second followed by (−3) times the firstrow added to the last. This yields 1 2 2 | 1 0 0

0 −2 0 | −1 1 00 −5 −7 | −3 0 1

 .

Then take 5 times the second row and add to -2 times the last row. 1 2 2 | 1 0 00 −10 0 | −5 5 00 0 14 | 1 5 −2

Next take the last row and add to (−7) times the top row. This yields −7 −14 0 | −6 5 −2

0 −10 0 | −5 5 00 0 14 | 1 5 −2

 .

Now take (−7/5) times the second row and add to the top. −7 0 0 | 1 −2 −20 −10 0 | −5 5 00 0 14 | 1 5 −2

 .

Finally divide the top row by -7, the second row by -10 and the bottom row by 14 whichyields 

1 0 0 | − 17

27

27

0 1 0 | 12 − 1

2 0

0 0 1 | 114

514 − 1

7

 .

Therefore, the inverse is − 1

727

27

12 − 1

2 0

114

514 − 1

7

