# characteristic equation

Exam/Review for Ch 5 and 6
You need to submit solution on Monday for
Name___________________________________Date:_____________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
For the given matrix and eigenvalue, find an eigenvector corresponding to the eigenvalue.
1) A = –24 14
84 46
, Ώ
= 4 1)
A)
1
2
B)
10
C)
2
1
D)
1
46
For the given matrix A, find a basis for the corresponding eigenspace for the given eigenvalue.
2) A =
1 4 4
4 1 4
4
4 7
, Ώ
= –3 2)
A)
101
,
01
1
B)
10
1
C)
01
1
D)
10
1
,
011
Find the characteristic equation of the given matrix.
3) A =
1 7 4 9
0
5 7 1
0 0
7 5
0 0 0 6
3)
A) (6
Ώ)(5 Ώ)(1 Ώ)(9 Ώ) = 0 B) (1 Ώ)(5 Ώ)(7 Ώ)(6 Ώ) = 0
C) (9
Ώ)(1 Ώ)(5 Ώ)(6 Ώ) = 0 D) (1 Ώ)(7 Ώ)(4 Ώ)(9 Ώ) = 0
Find the eigenvalues of the given matrix.
4) 0 1
2 3
4)
A)
2 B) 1 C) 1, 2 D) 1, 2
The characteristic polynomial of a 5 × 5 matrix is given below. Find the eigenvalues and their multiplicities.
5) Ώ5 + 17Ώ4 + 72Ώ3 5)
A) 0 (multiplicity 3), 8 (multiplicity 1), 9 (multiplicity 1)
B)
9 (multiplicity 1), 8 (multiplicity 1)
C) 0 (multiplicity 1),
– –
D) 8 (multiplicity 1), 9 (multiplicity 1)
1

Find a formula for Ak, given that A = PDP1, where P and D are given below.
6) A = 5 3
2 10
, P
= 3 1
2 1
, D
= 7 0
0 8
6)
A)
3
· 7k 2 · 8k 3 · 8k 3 · 7k
2
· 7k 2 · 8k 3 · 8k 2 · 7k
B)
3
· 7k + 2 · 8k 3 · 8k + 3 · 7k
2
· 7k + 2 · 8k 3 · 8k + 2 · 7k
C)
3
· 7k 2 · 8k 3 · 8k + 3 · 7k
2
· 7k + 2 · 8k 3 · 8k 2 · 7k
D)
7k 0
0 8k
Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A= PDP1.
7) A =
1 1 4
0
4 0
5 1 8
7)
A)
P
=
1 0 1
0
4 0
1 1 1
, D
=
4 0 0
0 1 0
0 0
3
B)
P
=
1 0 1
9 4 0
1 1 1
, D
=
4 0 3
0
4 0
0
4 3
C)
P
=
1 9 1
9 4 0
1
4 1
, D
=
4 1 0
0
4 0
0 0
3
D)
P
=
1 0 1
9 4 0
1 1 1
, D
=
4 0 0
0
4 0
0 0
3
Find the matrix of the linear transformation T: V  W relative to B and C.
8) Suppose B = {b1, b2} is a basis for V and C = {c1, c2, c3} is a basis for W. Let T be defined by
T(
b1) = –5c1 6c2 + 5c3
T(
b2) = –5c1 12c2 + 2c3
8)
A)
5 0
6 6
5
3
B)
5 6 5
0 6 3
C)
5 6 5
5 12 2
D)
5 5
6 12
5 2
Define T: R2  R2 by T(x) = Ax, where A is the matrix defined below. Find the requested basis B for R2 and the
corresponding B
matrix for T.
9) Find a basis B for R2 and the Bmatrix D for T with the property that D is a diagonal matrix.
A
= –67 60
72 65
9)
A)
B
= 1
1
,
5
6
, D
= –7 0
0 5
B)
B
= 1
5
,
1
6
, D
= –7 0
0 5
C)
B
= 1
1
,
56
, D
= –7 0
0 5
D)
B
= 5
6
,
1
1
, D
= –7 0
0 5
2

Compute the dot product u · v.
10) u =
1
3
3
,
v =
5
2
3
10)
A) 8 B)
8 C) 0 D) 2
Find a unit vector in the direction of the given vector.
11) 16
32
11)
A)
1
3

2
3
B)
15

25
C)
1
5

2
5
D)
1
5
2
5
Determine whether the set of vectors is orthogonal.
12)
363
,
3
0
3
,
3
3
3
12)
A) Yes B) No
13)
20
40
20
,
20
0
20
,
20
20
20
13)
A) Yes B) No
Find the distance between the two vectors.
14) u = (0, 0, 0) , v = (6, 9, 9) 14)
A) 24 B) 2 6 C) 3 22 D) 198
Express the vector x as a linear combination of the u’s.
15) u1 =
2
0
1
,
u2 =
356
,
u3 =
2
6
4
,
x =
4
14
33
15)
A)
x = –5u1 + 4u2 + 4u3 B) x = 5u1 + 2u2 4u3
C)
x = –5u1 2u2 + 4u3 D) x = 10u1 + 4u2 8u3
Find the orthogonal projection of y onto u.
16) y = –3
4
,
u = 5
10
16)
A)
1
2
B)
5
10
C)
25
50
D)
15

25
3

Let W be the subspace spanned by the u’s. Write y as the sum of a vector in W and a vector orthogonal to W.
17) y =
12
14
25
,
u1 =
2
2
1
,
u2 =
1
3
4
17)
A)
y =
1
21
17
+
11
7
8
B)
y =
2
42
34
+
10
28
9
C)
y =
1
21
17
+
11
7
8
D)
y =
1
21
17
+
13
35
42
Find the new coordinate vector for the vector x after performing the specified change of basis.
18) Consider two bases B = b1, b2 and C = c1, c2 for a vector space V such that
b1 = c1 6c2 and b2 = 4c1 + 3c2. Suppose x = b1 + 2b2. That is, suppose [x]B = 1
2
. Find [
x]C.
18)
A)
6
9
B)
90
C)
80
D)
11
10
Find the specified changeofcoordinates matrix.
19) Consider two bases B = b1, b2 and C = c1, c2 for a vector space V such that
b1 = c1 2c2 and b2 = 3c1 4c2. Find the changeofcoordinates matrix from B to C.
19)
A)
1 3
2 4
B)
1 3
2 4
C)
1
2
3
4
D)
0 3
2 4
Find the closest point to y in the subspace W spanned by u1 and u2.
20) y =
10
20
33
,
u1 =
2
2
1
,
u2 =
1
3
4
20)
A)
1
27
25
B)
11
23
5
C)
8
96
95
D)
1
27
25
4