characteristic equation

Exam/Review for Ch 5 and 6
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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 = PDP–1, 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= PDP–1.
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 B–matrix 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 change–of–coordinates 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 change–of–coordinates 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