240 Higher Engineering Mathematics
The determinant of
⎛
⎝
1
5 −2
3 −1
4
−3
6 −7
⎞
⎠
= 1(7 − 24) − 5(−21 + 12) − 2(18 − 3)
= −17 + 45 − 30 = −2
Hence the inverse of
⎛
⎝
1
5 −2
3 −1
4
−3
6 −7
⎞
⎠
=
⎛
⎝
−17
23
18
9 −13 −10
15 −21 −16
⎞
⎠
−2
=
⎛
⎝
8.5 −11.5 −9
−4.5
6.5
5
−7.5
10.5
8
⎞
⎠
Now try the following exercise
Exercise 97 Further problems on the
inverse of a 3 by 3 matrix
1. Write down the transpose of
⎛
⎝
4 −7
6
−2
4
0
5
7 −4
⎞
⎠
⎡
⎣
⎛
⎝
4 −2
5
−7
4
7
6
0 −4
⎞
⎠
⎤
⎦
2. Write down the transpose of
⎛
⎝
3
6
1
2
5 −
2
3 7
−1
0
3
5
⎞
⎠
⎡
⎣
⎛
⎝
3
5 −1
6 −
2
3
0
1
2
7
3
5
⎞
⎠
⎤
⎦
3. Determine the adjoint of
⎛
⎝
4 −7
6
−2
4
0
5
7 −4
⎞
⎠
⎡
⎣
⎛
⎝
−16
14 −24
−8 −46 −12
−34 −63
2
⎞
⎠
⎤
⎦
4. Determine the adjoint of
⎛
⎜
⎝
3
6
1
2
5 −
2
3 7
−1
0
3
5
⎞
⎟
⎠
⎡
⎢
⎢
⎣
⎛
⎜
⎜
⎝
−
2
5 −3
3
5
42
1
3
−10 2
3
10 −18
1
2
−
2
3
−6 −32
⎞
⎟
⎟
⎠
⎤
⎥
⎥
⎦
5. Find the inverse of
⎛
⎝
4 −7
6
−2
4
0
5
7 −4
⎞
⎠
⎡
⎣ −
1
212
⎛
⎝
−16
14 −24
−8 −46 −12
−34 −63
2
⎞
⎠
⎤
⎦
6. Find the inverse of
⎛
⎜
⎝
3
6
1
2
5 −
2
3 7
−1
0
3
5
⎞
⎟
⎠
⎡
⎢
⎣−
15
923
⎛
⎜
⎝
−
2
5 −3
3
5
42
1
3
−10 2
3
10 −18
1
2
−
2
3
−6 −32
⎞
⎟
⎠
⎤
⎥
⎦