426 VECTORS AND MATRICES [APP. A
(c) Computef (A)by first substitutingAforxand 5Ifor the constant term 5 inf(x)= 2 x^3 − 4 x+5:f (A)= 2 A^3 − 4 A+ 5 I= 2[
− 730
60 − 67]
− 4[
12
4 − 3]
+ 5[
10
01]Then multiply each matrix by its respective scalar:f (A)=[
− 14 60
120 − 134]
+[
− 4 − 8
−16 12]
+[
50
05]Lastly, add the corresponding elements in the matrices:f (A)=[
− 14 − 4 + 560 − 8 + 0
120 − 16 + 0 − 134 + 12 + 5]
=[
− 13 52
104 − 117](d) Computeg(A)by first substitutingAforxand 11Ifor the constant term 11 ing(x)=x^2 + 2 x−11:g(A)=A^2 + 2 A− 11 I=[
9 − 4
− 817]
+ 2[
12
4 − 3]
− 11[
10
01]=[
9 − 4
− 817]
+[
24
8 − 6]
+[
− 11 0
0 − 11]
=[
00
00](Sinceg(A)=0, the matrixAis a zero of the polynomialg(x).)A.15. Compute each determinant: (a)
∣
∣
∣
∣45
− 3 − 2∣
∣
∣
∣;(b)∣
∣
∣
∣a−bb
ba+b∣
∣
∣
∣.(a)∣
∣∣
∣45
− 3 − 2∣
∣∣
∣=^4 (−^2 )−(−^3 )(^5 )=−^8 +^15 =^7.(b)∣∣
∣∣a−bb
ba+b∣∣
∣∣=(a−b)(a+b)−b^2 =a^2 −b^2 −b^2 =a^2 − 2 b^2.A.16. Find the determinant of each matrix:
(a) A=⎡
⎣123
4 − 23
05 − 1⎤
⎦; (b) B=⎡
⎣4 − 1 − 2
02 − 3
521⎤
⎦; (c) C=⎡
⎣2 − 34
12 − 3
− 1 − 25⎤
⎦(Hint: Use the diagram in Fig. A-3 (b)):(a) |A|= 2 + 0 + 60 − 0 − 15 + 8 = 55
(b) |B|= 8 + 15 + 0 + 20 + 24 + 0 = 67
(c) |C|= 20 − 9 − 8 + 8 − 12 + 15 = 14A.17. Find the inverse of: (a)A=
[
53
42]
;(b)B=[
− 26
3 − 9]
.Use the formula in Section A.9.(a) First find|A|= 5 ( 2 )− 3 ( 4 )= 10 − 12 =− 2 .Next, interchange the diagonal elements, take the negatives of the
nondiagonal elements, and multiply by 1/|A|:A−^1 =−
1
2[
2 − 3
− 45]
=[− (^132)
2 −^52
]
(b) First find|B|=− 2 (− 9 )− 6 ( 3 )= 18 − 18 = 0 .Since|B|= 0 ,Bhas no inverse.
A.18. Find the inverse of: (a)A=
⎡
⎣
1 − 22
2 − 36
117
⎤
⎦;(b)B=
⎡
⎣
13 − 4
15 − 1
313 − 6
⎤
⎦.