Fundamentals of Matrix Algebra, 2011a

Chapter A
Soluons To Selected Problems
⎡
⎤
(b) ⃗x = ⎣ 4 0 ⎦
−2
11. (a) det (A) = 0, det (A 1 ) = 147,
det (A 2 ) = −49, det (A 3 ) = −49
(b) No soluon exists.
Chapter 4
Secon 4.1
1. λ = 3
3. λ = 0
5. λ = 3
[ ] −1
7. ⃗x =
2
⎡
9. ⃗x = ⎣ 3
⎤
−7 ⎦
7
⎡
11. ⃗x = ⎣ −1
⎤
1 ⎦
1
[ ] 9
13. λ 1 = 4 with ⃗x 1 = ;
1
[ ] 8
λ 2 = 5 with ⃗x 2 =
1
[ ] −2
15. λ 1 = −3 with ⃗x 1 = ;
1
[ ] 6
λ 2 = 5 with ⃗x 2 =
1
[ ] 1
17. λ 1 = 2 with ⃗x 1 = ;
1
[ ] −1
λ 2 = 4 with ⃗x 2 =
1
[ ] 1
19. λ 1 = −1 with ⃗x 1 = ;
2
[ ] 1
λ 2 = −3 with ⃗x 2 =
0
⎡
21. λ 1 = 3 with ⃗x 1 = ⎣ −3
⎤
0 ⎦;
2
⎡
λ 2 = 4 with ⃗x 2 = ⎣ −5
⎤
−1 ⎦
1
⎡ ⎤
λ 3 = 5 with ⃗x 3 = ⎣ 1 0 ⎦
0
⎡
23. λ 1 = −5 with ⃗x 1 = ⎣ 24
⎤
13 ⎦;
8
⎡ ⎤
λ 2 = −2 with ⃗x 2 = ⎣ 6 5 ⎦
1
⎡ ⎤
λ 3 = 3 with ⃗x 3 = ⎣ 0 1 ⎦
0
⎡ ⎤
25. λ 1 = −2 with ⃗x 1 = ⎣ 0 0 ⎦;
1
⎡ ⎤
λ 2 = 1 with ⃗x 2 = ⎣ 0 3 ⎦
5
⎡
λ 3 = 5 with ⃗x 3 = ⎣ 28
⎤
7 ⎦
1
⎡ ⎤
27. λ 1 = −2 with ⃗x = ⎣ 1 0 ⎦;
1
⎡ ⎤
Secon 4.2
λ 2 = 3 with ⃗x = ⎣ 1 1 ⎦;
1
⎡ ⎤
λ 3 = 5 with ⃗x = ⎣ 0 1 ⎦
1
[ ] 4
1. (a) λ 1 = 1 with ⃗x 1 = ;
1
[ ] 1
λ 2 = 4 with ⃗x 2 =
1
[ ] −1
(b) λ 1 = 1 with ⃗x 1 = ;
1
[ ] −1
λ 2 = 4 with ⃗x 2 =
4
[ ] 1
(c) λ 1 = 1/4 with ⃗x 1 = ;
1
[ ] 4
λ 2 = 4 with ⃗x 2 =
1
(d) 5
(e) 4
[ ] −5
3. (a) λ 1 = −1 with ⃗x 1 = ;
1
[ ] −6
λ 2 = 0 with ⃗x 2 =
1
[ ] 1
(b) λ 1 = −1 with ⃗x 1 = ;
6
[ ] 1
λ 2 = 0 with ⃗x 2 =
5
(c) Ais not inverble.
234