Math 291 - Practice Problems for Final
These problems are in addition to the Practice Problems for Tests 1 and 2.
1. Find the eigenvalues for each of the following matrices:
2 sin θ − sin θ
a a2
B=
A=
sin θ
0
− a2 0
0 1
2. For what values of a does the matrix A =
have the following characteristics?
a 1
(a) A has an eigenvalue of multiplicity 2.
(b) A has −1 and 2 as eigenvalues.
(c) A has real eigenvalues.


2 0 0
3. Let A = 1 2 1.
0 1 2
(a) Find the characteristic equation of A.
(b) Find the eigenvalues of A.
(c) Find the eigenvectors corresponding to each eigenvalue.
(d) Find an invertible matrix P such that P −1 AP is diagonal.


1 2 3
4. Find the eigenvalues of the matrix 0 4 5 .
0 0 17


0 2 0
5. Find a diagonal matrix D that is similar to 2 0 4.
0 4 0
 


1
1 2 −2
7. Find the eigenvalue corresponding to the eigenvector 1 for the matrix −2 5 −2.
3
−6 6 −3
8. Give an example of a 2 × 2 matrix that is not diagonalizable.


0 1 1
9. Find an matrix P that diagonalizes the matrix A = 1 0 1.
1 1 0
10. For the subspace W = Span{(1, 1, 1)} of R3 , find a basis for W ⊥ .
11. Answer the following True or False:
1 3
(a) T F A =
is diagonalizable.
0 1
(b) T F Similar matrices have the same eigenvalues.
(c) T F If a square matrix A is diagonalizable then A has distinct eigenvalues.
(d) T F A square matrix A is symmetric if A = A−1 .
(e) T F A symmetric matrix can be diagonalized.
(f) T F A symmetric matrix has distinct eigenvalues.
(g) T F If W = Span{(1, 2)}, then W ⊥ = Span{(−2, 1)}.
(h) T F Every diagonal matrix is invertible.
(i) T F A symmetric matrix has an inverse iff it has nonzero eigenvalues.
(l) T F If λ is an eigenvalue of a matrix, then there is a unique eigenvector of the matrix that
corresponds to λ.
(m) T F The row space of any matrix equals the row space of its reduced row echelon form.
12. Let (i) a0 = 1, a1 = 1, ak+1 = 3ak + 4ak−1 for k ≥ 1 and (ii) a0 = 0, a1 = 1, ak+1 =
2ak + 3ak−1 , for k ≥ 1.
(a) List the first 5 terms of the sequence:
(b) Find the 2 × 2 matrix A such that xk+1 = Axk : A =
(c) Find the eigenvalues and corresponding eigenvectors for A.
(c) Find a matrix P such that P −1 AP is a diagonal matrix.
(e) Use matrices to find a formula for ak . ak =
13. The following vectors form a basis for R2 . Use the Gram-Schmidt process to construct an
orthonormal basis for R2 .
1
6
,
1
2
14. The following vectors form a basis for R3 . Use the Gram-Schmidt process to construct an
orthonormal basis for R3 .
     
2
2 
 1
1 , 0 , 4


1
1
5
15. Each month U-Haul trucks are driven among the cities of Denver, St. Louis and Kansas City.
Half of the trucks in Denver stay in Denver, with the remaining trucks split evenly between St.
Louis and Kansas City.
1/3 of the trucks in St. Louis remain in St. Louis while 1/6 of the trucks go to Kansas City and
1/2 of the trucks go to Denver.
In Kansas City 1/5 of the trucks stay in Kansas City, 1/5 go to St. Louis and 3/5 go to Denver.
(a) Let D=Denver, S=St. Louis, and K=Kansas City. Below is the stochastic transition matrix P
representing this system. Fill in the remaining entries of the matrix.
1
2



P =






1
3
D




 S



1 
K
5
(b) If there are initially 200 trucks in Denver, 300 trucks in St. Louis, and 400 trucks in Kansas
City, how many trucks are in Denver after one month?
(c) How many trucks are in Denver after one year?
(d) Why is P a regular stochastic matrix? Find the steady-state vector for P .
3 5
16. Consider the Leslie matrix L =
.
.8 0
(a) Find the eigenvalues and eigenvectors for L.
(b) Find the growth rate λ1 (the dominant eigenvalue from (a)).
(c) Find the stable range distribution vector x1 (x1 is the eigenvector corresponding to λ1 with 1 as
the leading component).For
100
(d) For N (0) =
(the initial population vector), find c such that for large k, N (k) ≈ (λ1 )k cx1
100
(use k = 10 to estimate c).
17. Let 1 ≤ i < j ≤ n and E1 , E2 , E3 ∈ Mnn where E1 is the identity matrix with rows i, j
interchanged, E2 is the identity matrix with the ith row multiplied by a nonzero scalar c, and E3 is
the identity matrix with the ith row multiplied by a scalar c and added to the jth row.
(a) Compute |E1 |, |E2 |, |E3 |.
(b) E1T , E2T , E3T are also elementary matrices. Describe each one in the same manner as E1 , E2 , E3
(fill in the blank).
are described, i.e., the identity matrix with
Answers to Practice Problems for the Final
1. Eigenvalues for A are λ =
multiplicity 2.
2 (a) a = − 41
(b) a = 2
a
2
with multiplicity 2.
Eigenvalues for B are λ = sin θ with
(c) a ≥ − 41
3 (a) (λ − 2)(λ − 3)(λ − 1) = 0 (b) λ = 1, 2, 3
 
 
 
0
−1
0
(c) λ = 1 : t −1 λ = 2 : t  0 λ = 3 : t 1
1
1
1


0 −1 0
0 1
(d) P = −1
1
1 1


0 √0
0

20
4. λ = 1, 4, 17 5. 0
√0
0 0 − 20


1 −1 −1
1 2
0
7. λ = −3 8.
9. 1 1
0 1
1 0
1
(λ = −1, −1, 2)
10. (a, b, c) ∈ W ⊥ iff (a, b, c) · (1, 1, 1) = a + b + c = 0 iff c = −a − b iff (a, b, −a − b) ∈ W ⊥ .
A basis for W ⊥ is {(1, 0, −1), (0, 1, −1)}.
−1 0
(e) T
(f) F (g) T (h) F (i) T
11. (a) F (b) T (c) F (d) F
1 1
(l) F (m) T
0 1
1
−1
1 −1
12. (i) (a) 1,1,7,25,103 (b) A =
(c) λ = 4,
and λ = −1,
(d) P =
(e)
4 3
4
1
4 1
3(−1)k + 2 · 4k
ak =
5
0 1
1
−1
1 −1
(ii) (a) 0,1,2,7,20 (b) A =
(c) λ = 3,
and λ = −1,
(d) P =
(e)
3 2
3
1
3 1
(−1)k+1 + 3k
ak =
4
1
1 1
1
1
1
13. Orthogonal basis
,
. Orthonormal basis √
,√
.
1
−1
2 1
2 −1
     

 
 
 
1
1 
1
1
 1
 1 1
1   1  








1 , −1 , 1
1
14. Orthogonal basis
. Orthonormal basis √ 1 , √ −1 , √


 3

2
6
1
0
−2
1
0
−2


1/2 1/2 3/5
15. (a) P = 1/4 1/3 1/5 where the first column is Denver, second column is St. Louis, and
1/4 1/6 1/5
third column is Kansas city.
   
200
490



(b) P 300 = 230 There are 490 trucks in Denver at the end of one month.
400
  180


200
469.57
(c) P 12 300 = 234.78. There are approximately 469-470 trucks in Denver after one year.
400
195.65
(d) P is a regular stochastic matrix as all of the entries are positive and each column adds to 1. To
solve P q = q, we convert to the homogeneous equation (I −P )q = 0 and solve for the NullSp(I-P).


 


1
0 −2.4
2.4
1/2 −1/2 −3/5


I −P = −1/4 2/3 −1/5 rref
 0 1 −1.2. The NullSp(I-P) = Span{1.2}.
−−→
1
−1/4 −1/6 4/5
0 0
0
 
2.4
1

1.2.
The steady state vector is a probability vector q =
2.4 + 1.2 + 1
1
16. (a)
Eigenvalues are 4, −1. An eigenvector corresponding
to the dominant eigenvector λ1 = 4
5
−5
is
and an eigenvector corresponding to λ = −1 is
.
1
4
(b) The growth rate is λ1 = 4.
1
(c) The stable range distribution vector is x1 =
.2
188743600
188743600
10
10 1
10
≈ 180. So, for large k,
(d) L N (0) =
≈ cλ1 x1 = c4
and c ≈
.2
410
37748800
180
N (k) ≈ 4k
.
36
17. (a) |E1T | = −1, |E2 | = c, |E3 | = 1. (b) E1T = E1 , E2T = E2 , E3 is the identity matrix with the
jth row multiplied by a scalar c and added to the ith row.