CHAPTER 1: MATRIX ALGEBRA
Matrix operations
1. Let
2
A
4
1
C
 2
0 1
7 5 1 
,
B

1 4 3
5 2 


2
 3 5
 5
,D
,E 


1
 1 4
3
Compute
a. 2 A, B  2 A, AC, CD.
b. A  3B, 2C  3E, DB, EC .
If an expression is undefined, explain why.
2. Compute A  5I 3 and (5 A) I 3 , where
 5 1 3 
A   4 3 6 
 3 1 2 
 2 1 1 
 2 1 0 
3. Let A  
and B  

 . Compute
0
1

4

3
2
2




a. 3A  2B
b.4 A  3B
c. AT .B, BT . A
4. Compute
 1 3 3  1 4 5 

a.  3 4 1 
 0 2 7 
 2 5 3  3 2 1 



7
 5 0 2 3 
3
b.  4 1 5 3   
 3 1 1 2   2 

 1 
 
5. If a matrix A is 5  3 and the product AB is 5  7 , what is the size of B?
6. How many rows does B have if BC is a 5  4 matrix?
 3 6
 1 1 
 3 5
7. Let A  
, B
and C  


 . Verify that AB  AC and yet
 1 2 
 3 4
2 1
B C.
 2 3
 1 9
8. Let A  
and B  

 . What value(s) of k , if any, will make AB  BA?

1
1

3
k




9. Find A n where n is a natural number and
 2 1 
a. A  

 3 2 
1  
b. A  

0 1 
 1 1 1
c. A   0 1 1
 0 0 1


a 1 0
10. Let A   0 a 1  . Find A2020 .
0 0 a


The rank of a matrix
11. Determine the rank of the following matrices
2 0
 1 1 3 
 1 10 8 

1 2




a.  2 1 3  b.  2 3 5  c. 
 3 2
 3 1 2
 3 7 3 





 5 2
3

5
e. 
1

7
1 3 2
3 2 3
3 5 0
5 1 4
2
1
5


1
4
f. 
7
1

1
1

1
3 1 
 2 1 3 2 

2 3
d.  4 2 5 1 
5 4 
 2 1 1 8 



8 5
1 1 1
3 1 1 
1 4 1

1 1 5
2 3 4

1 1 1 
12. Argue based on m the ranks of the following matrices
m
1
2 
1
 1 2 1 4 2 


2
m4 

 , B=  2 3m  1
A   2 1 1 1 1 
 4 5m  1 m  4 2 m  7 
 1 7 4 11 m 




2m
2
4 
2
 m 2 2 2
 1 2



2 m 2 2 
m 1

, D
C
 2 2 m 2 
1 m



 2 2 2 m 
1 2
1 1
1

1 1 1
0 1 1

2 1 1
The inverse of a matrix
13. Find the inverse of the following matrices (if exists)
 1 4 2
 1 1 2 
 2 1
 1 2 
3 2
 2 4 




A
, B  
, C  
, D  
, E   3 6 5, F   0 1 2
3 3 
 3 6
8 5
 4 6 
 2 2 3
0 0 1




14. Determine m such that the following matrices are invertible.
1
3
 m 1

a.  2
m  2 0
 2m
1
3 

 m m m
b.  1 m 1 
 1 1 m


1 2 
1
, B    . Find a matrix X such that AX  B .
15. Let A  

 5 12 
 5 
 1 2 
 3 0
, B
16. Let A  

 . Find a matrix X such that XA  B .
2 1 
 1 5 
17. Suppose P is invertible and A  PBP 1 . Solve for B in terms of A.
The determinant of a matrix
18. Compute
cos 2
ab c 1
a. 1 0 1 b. 4 2 3 c. b  c a 1 d. cos 2 
cos 2
1 1 0
2 3 6
ca b 1
0 1 1
1 1 1
1 2 3 4
e.
2
1 0 2
2 3 4 1
3 2 1 0
f.
3 4 1 2
1 0 1 3
4 1 2 3
1 2 1 3
a b
19. Suppose that d
e
g h
a
a. 3d
g
b
c
3e 3 f
h
i
g.
2 1 1
x
1 2 1
y
1 1 2
z
1 1 1
t
cos 2 
sin 2 
cos 2 
cos 2 
sin 2 
sin 2 
a b 0 1
h.
0 a b 1
1 0 a b
b 1 0 a
c
f  7 . Find the following determinants
i
g h
b. a b
d e
i
c
f
a
b
c
c. 2d  a 2e  b 2 f  c
g
h
i
20. Find x provided that
1 2 x 1 1
a.
c.
1
1 1
0
1 1
0 2
1
3
0
x
1
2
x
x 1 x
1 1
x
1 1
0
1 1
0 2
2
b.
1
1 x
0 1
0 2
1
1
x 1 1 1
1 1 x
 0 d.
x x 2 1
x x 1 3
1 1
1
(n  1)  x
21. Prove that
yz
zx
a. y1  z1
y2  z 2
z1  x1
z2  x2
x y
x
y
z
x1  y1  2 x1
x2  y2
x2
y1
y2
z1
z2
1 a a3
b. 1 b b3  (b  a)(c  a)(c  b)(a  b  c)
1 c
c3
0
CHAPTER 2: SYSTEM OF LINEAR EQUATIONS AND APPLICATIONS
System of linear equations
1. Solve the following systems
2 x  y  3z  9

a. 3x  5 y  z  4
4 x  7 y  z  5

3x  2 y  4 z  8

b. 2 x  4 y  5 z  11
4 x  3 y  2 z  1

 x1  x2  2 x3  1

c. 2 x1  x2  2 x3  4
4 x  x  4 x  2
3
 1 2
2 x1  2 x2  x3  19

d.  x1  2 x2  4 x3  31
4 x  6 x  9 x  2
2
3
 1
3x1  4 x2  x3  7

e.  x1  2 x2  3x3  0
7 x  10 x  5 x  2
2
3
 1
x  y  2z  0

f. 2 x  2 y  4 z  0
5 x  5 y  10 z  0

 x1  2 x2  2 x3  21

g. 5 x1  x2  2 x3  29
3x  x  x  10
 1 2 3
2. Solve the following systems
 x1  2 x2  3x3  2 x4  6

2 x1  x2  2 x3  3x4  8
a. 
3x1  2 x2  x3  2 x4  4
2 x1  3x2  2 x3  x4  8
2 x1  x2  3x3  2 x4  4

3x1  3x2  3x3  2 x4  6
b. 
3x1  x2  x3  2 x4  6
3x1  x2  3x3  x4  6
2 x1  2 x2  x3  x4  4

4 x1  3x2  x3  2 x4  6
c. 
8 x1  5 x2  3x3  4 x4  12
3x1  3x2  11x3  5 x4  6
3. Agrue solutions of the following systems based on the parameter m
mx  y  z  1

a.  x  my  z  1
 x  y  mz  1

mx  y  z  m

b. 2 x   m  1 y   m  1 z  m  1
 x  y  mz  1

4. Let
x  2 y  z  1

2 x   m  5  y  2 z  4

 x   m  3 y   m  1 z  m  3
a. Find m such that the system is inconsistent.
b. Find m such that the system has infinitely many solutions and verify the solution set.
5. Solve
 x1  x2  x3  x4  0

a.  x1  2 x2  x4  0
 x  x  3x  x  0
3
4
 1 2
2 x1  x2  4 x3  0

b. 3x1  5 x2  7 x3  0
4 x  5 x  6 x  0
2
3
 1
 x1  x2  x3  x4  0

c.  x1  2 x2  x4  0
 x  x  3x  x  0
3
4
 1 2
 x1  x2  5 x3  x4  0

 x1  x2  2 x3  3x4  0
d. 
3x1  x2  8 x3  x4  0
 x1  2 x2  9 x3  7 x4  0
6. Find a 
such that the system has nontrivial solution and determine the solution set.
a 2 x  3 y  2 z  0
2 x  y  z  0


a.  x  y  2 z  0 b. ax  y  z  0
8 x  y  4 z  0
5 x  y  az  0


Leontief Input – Output model
7. Solve the Leontief production equation for an economy with three sectors, given that
0, 4 0, 2 0,1 
C   0,1 0,3 0, 4 
0, 2 0, 2 0,3
 40 
D  110 
 40 
8. Solve the Leontief production equation for an economy with three sectors, given that:
 0,3 0,1 0,1 
A   0,1 0, 2 0,3 
0, 2 0,3 0, 2 
and the final demand for sectors are 70, 100 and 30 respectively.
9. Consider a three sector economy where the interindustry sales (sectors selling to each
other) and the final demand given in the following table:
Input (million dollars)
Output
Final
demand
1
2
3
1
20
60
10
50
2
50
10
80
10
3
40
30
20
40
a. Express the meaning of the figure 80.
b. Find the total output for each sector.
c. Find the input coefficients matrix A.
10. Consider a three sector economy where the interindustry sales (sectors selling to each
other) and the final demand given in the following table
Input
Output
Final demand
1
2
3
1
45
x
75
100
2
y
40
90
150
3
80
65
100
z
a. The total outputs are 310, 350 và 445 respectively. Determine x, y, z ?
b. Find the input coefficients matrix A.
CHAPTER 3: VECTOR SPACES
1. In
3
consider whether u is a linear combination of u1 , u2 , u3 .
a. u1   2,1,0 ; u2   3; 1;1 ; u3   2,0, 2 ; u  1,1,1
b. u1   2,4,3 ; u2  1, 1,0 ; u3  3,3,3 ; u   1,2,0 
2. Determine  such that u is a linear combination of u1 , u2 , u3 .
a. u1  1,2, 1 ; u2   2;1;3 ; u3   0,1, 1 ; u  1, ,2
b. u1  1, 2,3 ; u2   0, 1,   ; u3  1,0,1 ; u  3, 1,2 
3. Determine whether the following sets of vectors is linearly dependent or linearly
independent
a.
 2, 3,1 ;  3, 1,5 ; 1, 4,3 in
b.
 5, 4,3 ;  3,3, 2  ; 8,1,3 in
c.
 4, 5, 2,6  ;  2, 2,1,3 ;  6, 3,3,9  ;  4, 1,5,6  in
d.
1,0,0,0  ;  0,1,0,0  ;  0,0, a,0  in
3
3
4
4
provided that a 
4. Determine based on  whether the following sets of vectors is linearly dependent or
linearly independent

1 1
1

 1
 1 1 
v1    ,  ,   ; v2    ,  ,   ; v3    ,  ,   
2 2
2

 2
 2 2 

5. In
n
, suppose that u1; u2 ; u3 is linearly dependent. Prove that
v1  u2  u3 ; v2  u3  u1; v3  u1  u2 
is also linearly dependent.
CHAPTER 5: LINEAR TRANSFORMATION AND QUADRATIC FORMS
Eigenvalue and eigenvector
3 2 
1. Is   2 an eigenvalue of A  
 ? Why or why not?
3 8 
2. Find the eigenvalues of the matrices
0 0 0 
a. A   0 3 4 
 0 0 2 


 5 0 0
b. B   0 0 0 
 1 0 3 


3. Find the eigenvalues and eigenvectors of the following matrices
3 4 4
0 0 1 
 5 1 2 




A  1 1 3 ; B  1 0 0 ; C   1 5 2 
0 0 2
0 1 0
 2 2 2 
Diagonalization
4. Diagonalize
1 0 
a. A  

6 1
 2 1
b. A  

1 4 
5. Diagonalize the following matrices
1 2 2 
a. A   2 1 2 


 2 2 1 
 1 4 2 
b. A   3 4 0 


 3 1 3 
 2 0 1
d. A   0 2 1


 1 1 3 
1
0
e. A  
0

1
 5 1 2
c. A   1 5 2


 2 2 2 
0 0 0
0 0 0 
0 0 0

0 0 1
1 3
0 1
f. A  
0 0

0 0
2
1 3 
2 5

0 2 
1
 3 1 1 
6. Let A   1 5 1 . Compute An , n  .


 1 1 3 