MAT 121
1.
Sample Questions for Exam 1 – Matrix Algebra
 1 11
0
7
What is the size of the matrix product 
 3 4

3
2
a.
c.
3 3
43
2
1 0 3 1 
0  
0 2 4 0  ?

9
 0 1 0 1
0
b.
d.
4 4
3 4
4 
0 2  x   0

What value of x satisfies the matrix equation 

 ?
 4 3x  2   2  2 x 7 x  14
T
2.
a.
c.
3.
-4
4
b.
d.
-2
6
Solve for x and y in the following matrix equation:
1
0
3 2 2   x  y 3 2 6

1 0 1   4

x y  5 2 x  5 9

 
a.
c.
4.
( x, y)  (5,8)
( x, y)  (8,5)
b.
d.
( x, y)  (5,8)
( x, y)  (5,8)
 2  3
1  2 0
T
For matrices A  0 2  and B  
 , find the matrix 2( A  B ) , if possible.
5
1
2


7  2
a.
 6 4
 4 6


 14 0
d.
The matrix doesn’t exist.
b.
 5  1
 2 5 


 14 2 
c.
6  2 14
4 3 0 


5.
6.
Assuming all the operations given below are defined for matrices A , B and C , where
c is a scalar, which of the following properties is false?
 cAB 
c.
c  A  BT   cAT  B
d.
8.
T
T
b.
(cA  B)  C   c( AC )  BC
d.
 cAB 
3
 (c3 AB)( AB)2
 4 2 1 

2 
2 0 2 3   2 0
Find the matrix product 
, if possible.

1
0
1 3 0  1  5


2  1
0
a.
7.
  cBT  AT
a.
 12  1
4
b.
0 

 5 4 
The matrix product is undefined.
18 3 0
 0  1 7


c.
4  1
 18
 2  4 8 


2  3  4 3  1
Find the matrix product 
, if possible.

5  2 4 
0 1
2 4 
0  4


a.
4 3
6
  2  4 4


d.
The matrix product is undefined.
b.
c.
 4 7  1
 6 5 0 


2 3 
  1 0 6
For matrices A  
, B  2 3 and C  

 , which of the following
4  1
 3 4 5
operations cannot be performed?
a.
BAC
d.
 AB 
T
b.
T
e.
BC T  A
T
 BA  B 
c.
B( AT C )
9.
 5  2
Determine the inverse of 
 , if possible.
 7 3 
a.
 3 2
7 2 


d.
The inverse does not exist.
b.
3 2 
0  5


c.
 3 2
7 5 


Sample Problem 1
 0 3
 1 36 
3
For matrices A  
and B  
 , show that A  B  I , where I is the

48
1
4
0




2  2 identity matrix.
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Sample Problem 2
1 1 1
a b 0 


If the matrices 0 1 1 and  0 a b  constitute an inverse pair, then what are the values
0 0 1
 0 0 a 
of a and b ?
Sample Problem 3
A property of matrix algebra states that
 A  B 2
 A2  AB  BA  B2 ,
where A and B are both square matrices of the same size.
 1 1
3 2 
Illustrate the validity of this result using the matrices A  
and B  

.
  2 0
2  1
Sample Problem 4
Let A  1 0 1 .
(a)
Find the matrix product AT  A .
(b)
Show that AT  A  A  AT .
Bonus Problem
A matrix A is said to be upper-triangular if A is square and Aij  0 for all i  j .
What would be the minimum number of zero entries in a 5  5 upper-triangular matrix? Give
an example of such a matrix.
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