Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
For the given matrix A, find a basis for the corresponding eigenspace for the given eigenvalue.
1)
A =1– 4 – 4
– 4 1 4
4– 4 – 7 , = –3
1)
A)
0
1
–1
B)
1
0
1, 0
1
–1
C)
1
0
–1, 0
1
1
D)
1
0
–1
Find the eigenvalues of A, and find a basis for each eigenspace.
2)
A =2 4
–4 2
2)
A)
2– 4i, 1
–i ; 2+ 4i, 1
i
B)
2– 4i, 1
i ; 2+ 4i, 1
–i
C)
2– 4i, 1 + 2i
–4 ; 2+ 4i, 1 – 2i
–4
D)
2– 4i, 1 – 2i
–4 ; 2+ 4i, 1 + 2i
–4
1
Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A= PDP–1.
3)
A =
–8 0 0 0
0–8 0 0
1–4 8 0
–1 2 0 8
3)
A)
P =
16 32 0 0
8 8 0 0
1 0 1 0
0 1 0 1
, D =
–8 0 0 0
0–8 0 0
0 0 8 0
0 0 0 8
B)
Not diagonalizable
C)
P =
16 32 0 0
–8–8 0 0
1 0 1 0
0 1 0 1
, D =
8 0 0 0
0 8 0 0
0 0 –8 0
0 0 0 –8
D)
P =
16 –8 1 0
32 –8 0 0
0 0 1 0
0 0 0 1
, D =
8 0 0 0
0 8 0 0
0 0 –8 0
0 0 0 –8
Solve the initial value problem.
4)
x= Ax, x(0) =3
3.2 , where A =–4–3.125
8–4
4)
A)
x(t) =–2 sin 5t + 3 cos 5t
3.2 cos 5t + 4.8 sin 5t e–4t
B)
x(t) =2 sin 5t + –3 cos 5t
–3.2 cos 5t – 4.8 sin 5t e4t
C)
x(t) =–2 sin 5t + 3 cos 5t
3.2 cos 5t + 4.8 sin 5t e4t
D)
x(t) =2 sin 5t – 3 cos 5t
–3.2 cos 5t – 4.8 sin 5t e–4t
Consider the difference equation xk+1= Axk, where A has eigenvalues and corresponding eigenvectors v1, v2, and v3
given below. Find the general solution of this difference equation if x0 is given as below.
5)
1=1, 2=0.5, 3=0.4, v1=–6
6
1, v2=–3
1
–3, v3=1
–3
3, and x0=–47
29
3
5)
A)
xk=(1)kv1+ 5(0.5)kv2+ 4(0.4)kv3
B)
xk=(1)kv1+(0.5)kv2+(0.4)kv3
C)
xk=6(1)kv1+ 5(0.5)kv2+ 4(0.4)kv3
D)
xk=6(1)kv1+ 5(0.5)kv2+(0.4)kv3
2
Find the characteristic equation of the given matrix.
6)
A =
2 5 3 1
0 4 –5 8
0 0 8 7
0 0 0 2
6)
A)
(2 –)2(4 –)(8–) = 0
B)
(2 –)(7–)(8–)(1–) = 0
C)
(2 –)(4–)(8–) = 0
D)
(2 –)(5–)(3–)(1–) = 0
The characteristic polynomial of a 5 × 5 matrix is given below. Find the eigenvalues and their multiplicities.
7)
5– 134– 93+ 4052
7)
A)
0 (multiplicity 2), 9 (multiplicity 2), –5 (multiplicity 1)
B)
0 (multiplicity 2), –9 (multiplicity 2), –5 (multiplicity 1)
C)
0 (multiplicity 2), –9 (multiplicity 2), 5 (multiplicity 1)
D)
0 (multiplicity 1), 9 (multiplicity 3), –5 (multiplicity 1)
A
Determine whether the origin is an attractor, repellor, or a saddle point of the dynamical system xk+1= Axk, where A is
given below. Determine the direction of greatest attraction or repulsion, appropriately.
8)
A =0.9 –0.4
0.5 1.110223025e–16
8)
A)
Saddle point; direction of greatest attraction: along the line through 0 and –4
–5, direction of
greatest repulsion: along the line through 0 and 1
1
B)
Attractor; direction of greatest attraction: along the line through 0 and –4
–5
C)
Repellor; direction of greatest repulsion: along the line through 0 and 1
1
D)
Attractor; direction of greatest attraction: along the line through 0 and 1
1
D
3
A
For the given matrix and eigenvalue, find an eigenvector corresponding to the eigenvalue.
9)
A =–24 –14
84 46 , =4
9)
A)
1
–2
B)
1
46
C)
–2
1
D)
1
0
Determine whether the origin is an attractor, repellor, or a saddle point of the dynamical system xk+1= Axk, where A is
given below. Determine the direction of greatest attraction or repulsion, appropriately.
10)
A =5.0 0
01.2
10)
A)
Attractor; direction of greatest attraction: along the line through 0 and 1
0
B)
Attractor; direction of greatest repulsion: along the line through 0 and 0
1
C)
Saddle point; direction of greatest attraction: along the line through 0 and 1
0, direction of
greatest repulsion: along the line through 0 and 0
1
D)
Repellor; direction of greatest repulsion: along the line through 0 and 1
0
Apply the power method to the matrix A below with x0=0
1. Stop when k = 5, and determine the dominant eigenvalue
and corresponding eigenvector.
11)
A =–56 –15
220 59
11)
A)
4, 1
–4
B)
–1, 1
–4
C)
4, –3
11
D)
–1, –3
11
4
Find the matrix of the linear transformation T: V
W relative to B and C.
12)
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
12)
A)
–5–6 5
0 6 3
B)
–5–6 5
–5–12 2
C)
–5–5
–6–12
D)
–5 0
–6–6
For the given matrix A, find a basis for the corresponding eigenspace for the given eigenvalue.
13)
A =–4 0 0
2–6 0
–8–16 2, = –4
13)
A)
1
–1
–4
B)
1
1
0, 1
0
–4
C)
1
1
4
D)
1
1
0, 1
0
4
Find the eigenvalues of the given matrix.
14)
85 16
–440 –83
14)
A)
–3, 5
B)
5
C)
3, –5
D)
3
5
Use the inverse power method to determine the smallest eigenvalue of the matrix A.
15)
Assume that the eigenvalues are roughly 0.8, 2.6, and 16.
A =1 0 0
–0.75 1.75 0.75
–0.75 0.75 4.25
15)
A)
2.5
B)
1
C)
6
D)
0.8
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.
16)
Find a basis B for R2 and the B–matrix D for T with the property that D is a diagonal matrix.
A =–67 –60
72 65
16)
A)
B =1
5, –1
6, D =–7 0
0 5
B)
B =1
–1, 5
6, D =–7 0
0 5
C)
B =1
–1, 5
–6, D =–7 0
0 5
D)
B =5
–6, 1
–1, D =–7 0
0 5
6
Determine whether the origin is an attractor, repellor, or a saddle point of the dynamical system xk+1= Axk, where A is
given below. Determine the direction of greatest attraction or repulsion, appropriately.
17)
A =11 –7
10.5 –6.5
17)
A)
Saddle point; direction of greatest attraction: along the line through 0 and 1
1, direction of
greatest repulsion: along the line through 0 and –2
–3
B)
Saddle point; direction of greatest attraction: along the line through 0 and 2
3, direction of
greatest repulsion: along the line through 0 and 1
1
C)
Attractor; direction of greatest attraction: along the line through 0 and 1
1
D)
Repellor; direction of greatest repulsion: along the line through 0 and 2
3
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.
18)
Find a basis B for R2 and the B–matrix D for T with the property that D is an upper triangular
matrix.
A =–232 –1156
49 244
18)
A)
B =34
–5, –7
1, D =6 1
0 6
B)
B =34
7, 5
1, D =–6 1
0–6
C)
B =34
–7, –5
1, D =6 1
0 7
D)
B =34
–7, –5
1, D =6 1
0 6
7
Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A= PDP–1.
19)
A =6 0 0
1 6 0
0 0 6
19)
A)
P =1 0 0
6 6 0
0 1 1 , D =6 1 0
0 6 0
0 0 6
B)
Not diagonalizable
C)
P =1 6 1
0 6 1
–1 0 1 , D =6 0 0
0 6 0
0 0 6
D)
P =1 0 –1
6 6 0
1 1 1 , D =6 0 1
1 6 1
0 0 6
Find the eigenvalues of the given matrix.
20)
0–1
2 3
20)
A)
1, –2
B)
–2
C)
1
D)
1, 2
Find the matrix of the linear transformation T: V
W relative to B and C.
21)
Suppose B = {b1, b2, b3} is a basis for V and C = {c1, c2} is a basis for W. Let T be defined by
T(b1) =5c1+c2
T(b2) =6c1– 6c2
T(b3) =5c1– 6c2
21)
A)
5 1
6–6
5–6
B)
6 0 –1
1–6–6
C)
5 6
6 0
5–1
D)
5 6 5
1–6–6
8
Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A= PDP–1.
22)
A =
–3 0 0 0
0–3 0 0
– 12 3–912
0 0 0 –3
22)
A)
P =
2 0 –2 1
0 2 1 0
1 0 0 1
0 0 1 0
, D =
–3 0 0 0
0–3 0 0
0 0 –3 0
0 0 0 –9
B)
P =
4–2 1 0
8–2 0 0
1 0 1 1
0 1 1 0
, D =
–3 0 0 0
0–3 0 0
0 0 –9 0
0 0 0 –9
C)
Not diagonalizable
D)
P =
2 0 1 0
0 2 0 0
–2 1 0 1
1 0 1 0
, D =
–3 0 0 0
0–3 0 0
0 0 –3 0
0 0 0 –9
Find a formula for Ak, given that A = PDP–1, where P and D are given below.
23)
A =5 3
–210 , P =3 1
2 1 , D =7 0
0 8
23)
A)
7k 0
0 8k
B)
3·7k+2·8k3·8k+3·7k
2·7k+2·8k3·8k+2·7k
C)
3·7k– 2 ·8k3·8k+3·7k
2·7k+2·8k3·8k–2·7k
D)
3·7k– 2 ·8k3·8k–3·7k
2·7k–2·8k3·8k–2·7k
The characteristic polynomial of a 5 × 5 matrix is given below. Find the eigenvalues and their multiplicities.
24)
5+ 174+ 723
24)
A)
0 (multiplicity 3), 8 (multiplicity 1), 9 (multiplicity 1)
B)
0 (multiplicity 1), –9 (multiplicity 1), –8 (multiplicity 1)
C)
0 (multiplicity 1), 8 (multiplicity 1), 9 (multiplicity 1)
D)
0 (multiplicity 3), –9 (multiplicity 1), –8 (multiplicity 1)
9
Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A= PDP–1.
25)
A =1 1 4
0–4 0
–5–1–8
25)
A)
P =1 0 –1
–9–4 0
1 1 1 , D =–4 0 0
0–4 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 0 –1
0–4 0
1 1 1 , D =–4 0 0
0 1 0
0 0 –3
D)
P =1–9–1
–9–4 0
1–4 1 , D =–4 1 0
0–4 0
0 0 –3
Find the eigenvalues of A, and find a basis for each eigenspace.
26)
A =0.76 –1.04
0.64 0.44
26)
A)
–0.6 + 0.8i, 1 + 5i
4 ; –0.6 – 0.8i, 1 – 5i
4
B)
0.6 + 0.8i, 1 – 5i
4 ; 0.6 – 0.8i, 1 + 5i
4
C)
0.6 – 0.8i, 1 – 5i
4 ; 0.6 + 0.8i, 1 + 5i
4
D)
–0.6 – 0.8i, 1 + 5i
4 ; –0.6 + 0.8i, 1 – 5i
4
For the given matrix and eigenvalue, find an eigenvector corresponding to the eigenvalue.
27)
A =32 –9
108 –31 , =5
27)
A)
3
1
B)
1
–3
C)
1
3
D)
–3
1
10
Find the characteristic equation of the given matrix.
28)
A =
1–7 4 9
0–5 7 –1
0 0 –7 5
0 0 0 6
28)
A)
(1 –)(–5–)(–7–)(6–) = 0
B)
(1 –)(–7–)(4–)(9–) = 0
C)
(6 –)(5–)(–1–)(9–) = 0
D)
(9 –)(–1–)(5–)(6–) = 0
Apply the power method to the matrix A below with x0=0
1. Stop when k = 5, and determine the dominant eigenvalue
and corresponding eigenvector.
29)
A =–7–3
4 0
29)
A)
–4, 1
–1
B)
–3, 3
–4
C)
–4, 3
–4
D)
–3, 1
–1
A
11
A
Answer Key
Testname: C5
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