Fundamentals of Matrix Algebra, 2011a

(d) -1
(e) 0
⎡
5. (a) λ 1 = −4 with ⃗x 1 = ⎣ −7
⎤
−7 ⎦;
6
⎡ ⎤
λ 2 = 3 with ⃗x 2 = ⎣ 0 0 ⎦
1
⎡ ⎤
λ 3 = 4 with ⃗x 3 = ⎣ 9 1 ⎦
22
⎡
(b) λ 1 = −4 with ⃗x 1 = ⎣ −1
⎤
9 ⎦;
0
⎡
λ 2 = 3 with ⃗x 2 = ⎣ −20
⎤
26 ⎦
7
⎡
λ 3 = 4 with ⃗x 3 = ⎣ −1
⎤
1 ⎦
0
⎡
(c) λ 1 = −1/4 with ⃗x 1 = ⎣ −7
⎤
−7 ⎦;
6
⎡ ⎤
(d) 3
(e) −48
Chapter 5
Secon 5.1
λ 2 = 1/3 with ⃗x 2 = ⎣ 0 0 ⎦
1
⎡ ⎤
λ 3 = 1/4 with ⃗x 3 = ⎣ 9 1 ⎦
22
[ ] 1 2
1. A =
3 4
[ ] 1 2
3. A =
1 2
[ ] 5 2
5. A =
2 1
[ ] 0 1
7. A =
3 0
[ ] 0 −1
9. A =
−1 −1
11. Yes, these are the[ same; the transformaon
] −1 0
matrix in each is
.
0 −1
13. Yes, these are the same. [ Each produces ] the
1/2 0
transformaon matrix
.
0 3
Secon 5.2
1. Yes
3. No; cannot add a constant.
5. Yes.
⎡
7. [ T ] = ⎣ 1 2
⎤
3 −5 ⎦
0 2
⎡
9. [ T ] = ⎣ 1 0 3
⎤
1 0 −1 ⎦
1 0 1
11. [ T ] = [ 1 2 3 4 ]
Secon 5.3
⎡
1. ⃗x +⃗y = ⎣ 3 ⎤ ⎡
2 ⎦, ⃗x −⃗y = ⎣ −1
⎤
−4 ⎦
4
0
Sketches will vary slightly depending on
orientaon.
⃗x −⃗y
x
⃗x
⃗x +⃗y
⎡
3. ⃗x +⃗y = ⎣ 4 ⎤ ⎡
4 ⎦, ⃗x −⃗y = ⎣ −2
⎤
−2 ⎦
8
−4
z
⃗y
.
y
Sketches will vary slightly depending on
orientaon.
235