Matrices Word Problems

Matrices Word Problems
  • Page 1
 1.  
Solve system of equations. Determine whether the system is consistent and independent, consistent and dependent, or inconsistent by using augmented matrices.
2x + 5y = 11
4x - 3y = 9
a.
(1, 3)
b.
(3, 1)
c.
(- 1, 4)
d.
(1 2, 2)


Solution:

(25|114- 3|9)R1
R2

[Write the augmented matrix.]

Apply the row operation 1 / 2 × R1 to get 1 in the first row, first column.
(152|1124- 3|9)

Apply the row operation - 4 × R1 + R2 to get 0 in the second row, first column.
(152|1120- 13|- 13)

Apply the row operation - 1 / 13 × R2 to get 1 in the second row, second column.
(152|11201|1)

Apply the row operation -5 / 2 × R2 + R1 to get 0 in the first row, second column.
(10|301|1)

So, x = 3, y = 1
The solution is (3, 1).


Correct answer : (2)
 2.  
Choose the augmented matrix for the following system of equations.
6x + 4y - z = 4
x + y + z = 2
4x - y + 6z = 4
a.
(4- 16|4111|264- 1|4 )
b.
(111|464- 1|24- 16|4)
c.
(6 4- 1|4111|24- 16|4)
d.
(61- 1|4211|24- 16|4)


Solution:

The augmented matrix for the system of equations in the question is
(64- 1|4111|24- 16|4)


Correct answer : (3)
 3.  
Which one is the augmented matrix for the system of equations?
x - 2y + z = 4
5x - z = 4
y + 5z = - 7
a.
(11- 2|450- 1|4015|- 7)
b.
(1- 2105- 1015)
c.
(1- 21|450- 1|4015|- 7)
d.
(1- 2150- 1105)


Solution:

The augmented matrix for the system of equations is:
(1- 21|450- 1|4015|- 7)
[The coefficients are seperated from the constants by the vertical line segment.]


Correct answer : (3)
 4.  
Solve system of equations. Determine whether the system is consistent and independent, consistent and dependent, or inconsistent by using augmented matrices.
x - y + 3z = 5
4x + 2y - z = 0
x +3y + z = 5
a.
(10, - 10, - 5); consistent and independent
b.
(0, 1, 2); consistent and independent
c.
(10, - 10, - 5); consistent and dependent
d.
no solution; inconsistent


Solution:

(1-13|542-1|0131|5)R1
R2
R3

[Write the augmented matrix.]

Apply the row operation - 4×R1 + R2 to get 0 in the second row, first column.
(1-13|506-13|-20131|5)

Apply the row operation -1×R1 + R3 to get 0 in the third row, first column.
(1-13|506-13|-2004-2|0)

Apply the row operation 1 / 6×R2 to get 1 in the second row, second column.
(1-13|501-136|-20604-2|0)

Apply the row operation R2 + R1 to get 0 in the first row, second column.
(1056|10601-136|-20604-2|0)

Apply the row operation - 4×R2 + R3 to get 0 in the third row, second column.
(1056|10601-136|-20600203|403)

Apply the row operation 3 / 20×R3 to get 1 in the third row, third column.
(1056|10601-136|-206001|2)

Apply the row operation - 5 / 6×R3 + R1 to get 0 in the first row, third column.
(100|001-136|-206001|2)

Apply the row operation 13 / 6×R3 + R2 to get 0 in the second row, third column.
(100|0010|1001|2)

So, x = 0, y = 1, z = 2

So, the system is consistent and independent.


Correct answer : (2)
 5.  
Identify the augmented matrix for the system of equations.
x + z = 8
y - z = 8
x + y = 6
a.
(101|801- 1|- 8111|6)
b.
(101|801- 1|8110|6)
c.
(10101- 1110)
d.
(111|811- 1|8110|6)


Solution:

The augmented matrix for the system of equations given in the question is:
(101|801- 1|8110|6)
[The coefficients are separated from the constants by the vertical line segment.]


Correct answer : (2)
 6.  
Identify the augmented matrix for the system of equations.
32x+12y = z - 3
13x - y+12z = 6
x - 13z = - 6
a.
(3211|- 313- 112|610- 13|6)
b.
(3212- 1|- 313- 112|  610- 13|- 6)
c.
(310112)
d.
None of the above


Solution:

The augmented matrix for the system of equations given in the question is:
(3212- 1|- 313- 112|  610- 13|- 6)
[The coefficients are seperated from the constants by the vertical line segment.]


Correct answer : (2)
 7.  
Solve the system of equations.
x + z = 10
y + z = 7
x + y = 9
Determine whether the system is consistent and independent, consistent and dependent, or inconsistent by using augmented matrices.
a.
(6, 3, 4); consistent and independent
b.
(7, 2, 4); consistent and independent
c.
(7, 2, 4); consistent and dependent
d.
(6, 2, 5); inconsistent


Solution:

(101|10011|7110|9)R1
R2
R3

[Write the augmented matrix.]

Apply the row operation - 1 × R1 + R3 to get 0 in the third row, first column.
(101|10011|701- 1|-1)

Apply the row operation - 1 × R2 + R3 to get 0 in the third row, second column.
(101|10011|700- 2|-8)

Apply the row operation - 1 / 2 × R3 to get 1 in the third row, third column.
(101|10011|7001|4)

Apply the row operation - 1 × R3 + R1 to get 0 in the first row, third column.
(100|6011|7001|4)

Apply the row operation - 1 × R3 + R2 to get 0 in the second row, third column.
(100|6010|3001|4)

So, x = 6, y = 3, z = 4
The solution is (6, 3, 4).

So, the system is consistent and independent.


Correct answer : (1)
 8.  
Which of the following represents the system of equations for the augmented matrix?
(7- 1|   65   1|- 5)
a.
7x - y = - 6, 5x + y = - 5
b.
7x - y = 6, 5x + y = 5
c.
7x + y = 6, 5x - y = - 5
d.
7x - y = 6, 5x + y = - 5


Solution:

(7- 15   1) (xy) = (   6- 5)

(7x-y5x+y) = (   6- 5)
[Multiplication of matrices.]

7x - y = 6, 5x + y = - 5
[By the definition of equal matrices.]


Correct answer : (4)
 9.  
Find the system of equations represented by the augmented matrix.
(4- 19|- 4- 101|   49101|   8)


a.
IV
b.
II
c.
I
d.
III


Solution:

(4- 19|- 4- 101|   49101|   8)

(4x-y+10z- x+0y+z9x+10y+z) = (- 4   4   8)
[Multiplication of matrices.]

4x - y + 9z = - 4
- x + z = 4
9x + 10y + z = 8
[By the definition of equal matrices.]


Correct answer : (2)
 10.  
Find the number of solutions that the augmented matrix has. (100|2010|9001|7)
a.
infinite
b.
1
c.
2


Solution:

(100010001) (xyz) = (297)

(1x+0y+0z0x+1y+0z0x+0y+1z) = (297)
[Multiplication of matrices.]

(xyz) = (297)

x = 2, y = 9 & z = 7.
[By the definition of equal matrices.]

So, the number of solutions of the system of equations represented by the augmented matrix is one.


Correct answer : (3)

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