Name _______________________________
MAT 121 – Finite Mathematics
Professor Pestieau
Multiple-Choice Questions
February 14, 2013
Exam 1 – Matrix Algebra
[5 pts each]
Circle the correct answer for the questions provided below. Check your work to
minimize errors.
1.
 1 11
0
7
What is the size of the matrix product 
 3 4

3
2
2
4 0 0 
0  
 0 1 5  ?
9 
  3 2 1
0
a.
3 4
b.
4 4
c.
4 3
d.
3 3
x  0
7 x  14
 0
What value of x satisfies the matrix equation 
?


5 
3 x  2 5   x  8
T
2.
3.
a.
x4
b.
x  4
c.
x5
d.
x  2
Solve for x and y in the matrix equation below:
3 4 y   x  y 4 2  10 3 1  4 3 1


1 2 1   4
x 8  2 6 x 9  3 3x 0 

 
a.
(x, y)  (1,2)
b.
(x, y)  (1,2)
c.
(x, y)  (1,2)
d.
(x, y)  (1,2)




4.
5.
6.
2  3
1  2 0
For matrices A  0 2  and B  
, which of the following is the matrix
5 1 2

7  2
T
3( A  2 B) ?
a.
12 12 21
 21 12 6 


d.
This matrix is undefined.
b.
12 12 21
 21 12 6


c.
 4 7
 4 4 


 7 2 
Assuming all the operations given below are defined for matrices A , B and C , where
c and d are scalars, which of the following properties is false?
a.
c( AB  dB)  c  AB   cdB
b.
 cA  B 
T T
 cAT  B
c.
(cA  B)C   c( AC)  BC
d.
 cAB 
e.
c(dAB)T  cd ( BT AT )
3
 c( AB)( AB) 2
3 1
 1 3 6 0
Which of the following is the matrix product 0 4   
?

 2 1 0 1 


1 6 
a.
 5 8 18 1
 8 4 0 4 


 11 9 6 6 
b.
 5 8 2 3
 8 0 4 0 


 11 6 9 6 
c.
 5 18
 11 6 


d.
This matrix product is undefined.
7.
8.
1
T 3
 2 3 4  
Which of the following is the matrix product 
  2 4  ?

0 1 5  
 0 1 
4 3
6
  2  4 4


d.
This matrix product is undefined.
10.
b.
c.
 4 7  1
 6 5 0 


 1 0 
2 3 
Given matrices A  
, B  2 3 and C   3 4  , which of the following

4  1
 6 5 
operations cannot be performed?
a.
d.
9.
2 4 
 0  4


a.
CABT
All of the above.
b.
e.
 CB 
A
Only a and c.
T
c.
e.
C T (CAT ) BT
Only b and c.
 4 2 
Which of the following matrices is the inverse of 
?
 11 6 
a.
11 

 2 2 


 1 3
d.
The inverse does not exist.
b.
11 

3  2


 1 2 
c.
 3 1
 11


2
 2

Which of the following matrices is singular?
a.
1 3 
0 1


b.
 5 0 
 0 4


c.
 5 5 
 3 3 


d.
 1 2
 1 2


Show all your work on the following problems to receive full credit.
Problem 1
[15 pts]
0  1
 20 8
3
For matrices A  
and B  

 , show that A  B  I , where I is the 2  2
1
3
8
4




identity matrix.


Problem 2
[10 pts]
A matrix A is said to be symmetric if it is equal to its transpose (i.e. A  AT ).
a)

Explain why a symmetric matrix is necessarily square.

b)
Give an example of a 5  5 symmetric matrix with non-zero entries.

Problem 3
[10 pts]

 0
1 0 1 

1


If the matrices  2 2 1 and  
 2
 3 0 0 

 1

value of a ? Explain your work.
0

1
2
0
1 
a 

1 
constitute an inverse pair, then what is the
2 

1

a 
Problem 4
[15 pts]
To transpose the product of two matrices, we use the following property:
( AB) T  B T AT .
3 2 
Illustrate the validity of this property using the matrices A  1 2 and B  
.
2  1
Bonus Problem
[5 pts]
Using matrix algebra, show that the property given in Problem 4 can be extended to three matrices
as follows:
(ABC)T  CT BT AT .
