Question 1  
use Cramer’s Rule to solve the following system.
 3x - 4y = 4 
 2x + 2y = 12 
A. {(3, 1)} 
B. {(4, 2)} 
C. {(5, 1)} 
D. {(2, 1)}
Question 2  
Use Gaussian elimination to find the complete solution to each system.
 x1 + 4x2 + 3x3 - 6x4 = 5 
 x1 + 3x2 + x3 - 4x4 = 3 
 2x1 + 8x2 + 7x3 - 5x4 = 11 
 2x1 + 5x2 - 6x4 = 4 
A. {(-47t + 4, 12t, 7t + 1, t)} 
B. {(-37t + 2, 16t, -7t + 1, t)} 
C. {(-35t + 3, 16t, -6t + 1, t)} 
D. {(-27t + 2, 17t, -7t + 1, t)} 
 
Question 3  
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination. 
 x + y - z = -2 
 2x - y + z = 5 
 -x + 2y + 2z = 1 
A. {(0, -1, -2)} 
B. {(2, 0, 2)} 
C. {(1, -1, 2)} 
D. {(4, -1, 3)}
Question 4  
Use Gaussian elimination to find the complete solution to each system.
 x - 3y + z = 1 
 -2x + y + 3z = -7 
 x - 4y + 2z = 0 
A. {(2t + 4, t + 1, t)} 
B. {(2t + 5, t + 2, t)} 
C. {(1t + 3, t + 2, t)} 
D. {(3t + 3, t + 1, t)} 
Question 5  
Give the order of the following matrix; if A = [aij], identify a32 and a23. 


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1
0
-2

-5
7
  1/2


-6
  11

e
-∏
  -1/5

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A. 3 * 4; a32 = 1/45; a23 = 6 
B. 3 * 4; a32 = 1/2; a23 = -6 
C. 3 * 2; a32 = 1/3; a23 = -5 
D. 2 * 3; a32 = 1/4; a23 = 4 
 
Question 6  
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
 3x1 + 5x2 - 8x3 + 5x4 = -8
  x1 + 2x2 - 3x3 + x4 = -7 
 2x1 + 3x2 - 7x3 + 3x4 = -11 
 4x1 + 8x2 - 10x3+ 7x4 = -10 
A. {(1, -5, 3, 4)} 
B. {(2, -1, 3, 5)} 
C. {(1, 2, 3, 3)} 
D. {(2, -2, 3, 4)}
Question 7  
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination. 
x + 2y = z - 1 
 x = 4 + y - z 
 x + y - 3z = -2 
A. {(3, -1, 0)} 
B. {(2, -1, 0)} 
C. {(3, -2, 1)} 
D. {(2, -1, 1)} 
 
Question 8  
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination. 
x + 3y = 0 
 x + y + z = 1 
 3x - y - z = 11 
A. {(3, -1, -1)} 
B. {(2, -3, -1)} 
C. {(2, -2, -4)} 
D. {(2, 0, -1)} 
 
Question 9  
Use Cramer’s Rule to solve the following system.
 x + 2y + 2z = 5 
 2x + 4y + 7z = 19 
 -2x - 5y - 2z = 8 
A. {(33, -11, 4)} 
B. {(13, 12, -3)} 
C. {(23, -12, 3)} 
D. {(13, -14, 3)} 
 
Question 10  
Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists. 
 2w + x - y = 3 
 w - 3x + 2y = -4 
 3w + x - 3y + z = 1 
 w + 2x - 4y - z = -2 
A. {(1, 3, 2, 1)} 
B. {(1, 4, 3, -1)} 
C. {(1, 5, 1, 1)} 
D. {(-1, 2, -2, 1)} 
 

Question 11  

Solve the system using the inverse that is given for the coefficient matrix.

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2x + 6y + 6z = 8
2x + 7y + 6z =10
2x + 7y + 7z = 9


The inverse of:

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2
2
2

  6
7
7

  6
6
7

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is

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7/2
-1
0

  0
1
-1

  -3
0
1

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A. {(1, 2, -1)} 
B. {(2, 1, -1)} 
C. {(1, 2, 0)} 
D. {(1, 3, -1)} 
 
Question 12  
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination. 
 2x - y - z = 4 
 x + y - 5z = -4 
 x - 2y = 4 
A. {(2, -1, 1)} 
B. {(-2, -3, 0)} 
C. {(3, -1, 2)} 
D. {(3, -1, 0)} 
 Question 13  
Use Cramer’s Rule to solve the following system.
 2x = 3y + 2 
 5x = 51 - 4y 
A. {(8, 2)} 
B. {(3, -4)} 
C. {(2, 5)} 
D. {(7, 4)} 
 
Question 14  
Use Cramer’s Rule to solve the following system.
   12x + 3y = 15 
 2x - 3y = 13 
A. {(2, -3)} 
B. {(1, 3)} 
C. {(3, -5)} 
D. {(1, -7)} 
 
Question 15  
Use Cramer’s Rule to solve the following system.
x + y + z = 0 
 2x - y + z = -1 
 -x + 3y - z = -8 
A. {(-1, -3, 7)} 
B. {(-6, -2, 4)} 
C. {(-5, -2, 7)} 
 D. {(-4, -1, 7)} 
 
Question 16  
Use Cramer’s Rule to solve the following system.
4x - 5y - 6z = -1 
 x - 2y - 5z = -12 
 2x - y = 7 
A. {(2, -3, 4)} 
B. {(5, -7, 4)} 
C. {(3, -3, 3)} 
D. {(1, -3, 5)}

Question 17

 

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

A =

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0
0
1

1
0
  0

0
1
  0

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B =

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0
1
0

0
0
  1

1
0
  0

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A. AB = I; BA = I3; B = A 
B. AB = I3; BA = I3; B = A-1 
C. AB = I; AB = I3; B = A-1 
D. AB = I3; BA = I3; A = B-1

Question 18  
Use Cramer’s Rule to solve the following system.
   x + 2y = 3 
 3x - 4y = 4 
A. {(3, 1/5)} 
B. {(5, 1/3)} 
C. {(1, 1/2)} 
D. {(2, 1/2)} 
Question 19  
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination. 
 x + y + z = 4 
 x - y - z = 0 
 x - y + z = 2 
A. {(3, 1, 0)} 
B. {(2, 1, 1)} 
C. {(4, 2, 1)} 
D. {(2, 1, 0)} 
 
Question 20  
Use Cramer’s Rule to solve the following system.
   4x - 5y = 17 
 2x + 3y = 3 
A. {(3, -1)} 
B. {(2, -1)} 
C. {(3, -7)} 
 D. {(2, 0)} 

 

 

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