Page 280 - Advanced engineering mathematics

260 CHAPTER 8 Determinants

1 1 −2 1 1
b 22 = M 22 = = ,
120 120 2 −5 15

1 1 −2 1
b 23 =− M 32 =− = 0,
120 120 6 −3

1 1 63 2
b 31 = M 13 = = ,
120 120 29 5

1 1 −2 4 13
b 32 =− M 23 =− = ,
120 120 2 9 60

1 1 −24 1
b 33 = M 33 = =− .
120 120 6 3 4

Then
⎛ ⎞
1/10 29/120 −1/8
−1 ⎝ 1/5
B = A = 1/15 0 ⎠ .
2/5 13/60 −1/4

SECTION 8.4 PROBLEMS

⎛ ⎞
In each of Problems 1 through 10, test the matrix −14 1 −3
for singularity by evaluating its determinant. If the 6. ⎝ 2 −1 3 ⎠
matrix is nonsingular, use Theorem 8.4 to compute the 1 1 7
inverse. ⎛ 0 −4 3 ⎞
7. ⎝ 2 −1 6 ⎠

2 −1 1 −1 7
1.
1 6 ⎛ ⎞
11 0 −5
8. ⎝ 0 1
3 0 0 ⎠
2.
1 4 4 −7 9
⎛ ⎞
3 1 −2 1
−1 1
3. ⎜ 4 6 −3 9 ⎟
1 4 9. ⎜ ⎟
⎝ −2 1 7 4 ⎠

2 5 13 0 1 5
4.
−7 −3 ⎛ ⎞
7 −3 −4 1
⎛ ⎞
6 −1 3 10. ⎜ 8 2 0 0 ⎟
⎟
⎜
5. ⎝ 0 1 −4 ⎠ ⎝ 1 5 −1 7 ⎠
2 2 −3 3 −2 −5 9
8.5 Cramer’s Rule
Cramer’s rule is a determinant formula for the unique solution of a nonhomogeneous system
−1
AX = B when A is nonsingular. Of course, this is X = A B, but the following method is
sometimes convenient.

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October 14, 2010 14:26 THM/NEIL Page-260 27410_08_ch08_p247-266