A First Course in Linear Algebra, 2017a

132 Determinants
You can verify that AA −1 = I and hence our answer is correct.
♠
We will look at another example of how to use this formula to find A −1 .
Example 3.42: Find the Inverse From a Formula
Find the inverse of the matrix
⎡
A =
⎢
⎣
− 1 6
− 5 6
1
2
0
1
2
1
3
− 1 2
2
3
− 1 2
⎤
⎥
⎦
using the formula given in Theorem 3.40.
Solution. Firstweneedtofinddet(A). This step is left as an exercise and you should verify that det(A)=
1
6 . The inverse is therefore equal to A −1 = 1
(1/6) adj(A)=6adj(A)
We continue to calculate as follows. Here we show the 2×2 determinants needed to find the cofactors.
⎡
1
3
− 1 2
− 1 6
− 1 2
− 1 1
⎤T
6 3
2 3
− 1 −
2
− 5 6
− 1 2
− 5 2
6 3
1
A −1 0 1 1
1
= 6
−
2
2 2
2
0
2 3
− 1 2
− 5 6
− 1 −
2
− 5 2
6 3
⎢
1
0
1 1
1
2
2 2
2
0
⎥
⎣
1
∣ 3
− 1 −
2 ∣ ∣ − 1 6
− 1 2 ∣ ∣
− 1 1
⎦
6 3 ∣
Expanding all the 2 × 2 determinants, this yields
⎡
A −1 = 6
⎢
⎣
1
6
1
3
− 1 6
1
3
1
6
1
6
− 1 3
1
6
1
6
⎤
⎥
⎦
T
⎡
= ⎣
1 2 −1
2 1 1
1 −2 1
⎤
⎦
Again, you can always check your work by multiplying A −1 A and AA −1 and ensuring these products
equal I.
⎡
1 1
⎤
⎡
⎤ 2
0 2 ⎡ ⎤
1 2 −1
A −1 A = ⎣ 2 1 1 ⎦
− 1 1
6 3
− 1 1 0 0
2
⎢
⎥
1 −2 1 ⎣
⎦ = ⎣ 0 1 0 ⎦
0 0 1
− 5 6
2
3
− 1 2