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Cramer's Rule Worksheet

Cramer's Rule Worksheet
  • Page 1
 1.  
Evaluate the determinant.
|2223 |
a.
3
b.
2
c.
1
d.
4


Solution:

|2223 |

= 2(3) - 2(2) = 2.
[|abcd | = ad - bc.]


Correct answer : (2)
 2.  
Evaluate the determinant.
|4- 366 |
a.
42
b.
44
c.
- 44
d.
- 42


Solution:

|4- 366 |

= 4(6) - 6(- 3) = 42.
[|abcd | = ad - bc.]


Correct answer : (1)
 3.  
Evaluate.
|- 98- 32 |
a.
5
b.
6
c.
8
d.
7


Solution:

|- 98- 32 |

= - 9(2) - 8(- 3)
[|abcd | = ad - bc.]

= - 18 + 24 = 6


Correct answer : (2)
 4.  
Evaluate the determinant: |9020 |
a.
9
b.
12
c.
10


Solution:

|9020 |

= 9(0) - 2(0) = 0
[|abcd | = ad - bc.]


Correct answer : (2)
 5.  
Solve the system using Cramer's rule
x - y + 6 = - z
y + 1 = -z
x + 1 = z
a.
(- 2, 2, - 2)
b.
(- 3, 1, - 2)
c.
(- 4, - 1, - 3)
d.
(1, 3, - 4)


Solution:

(1-1101110-1)(xyz) = (-6-1-1)
[Write the matrix equation.]

D = |a1b1a2b2 | = |1-1101110-1 |
[Find D.]

D = 1 |110-1 | + 1 |011-1 | + 1 |0110 |

D = 1 [- 1 - 0] + 1[0 - 1] + 1[0 - 1] = - 1 - 1 - 1 = - 3

Dx = |c1b1c2b2 | = |-6-11-111-10-1 |

Dx = - 6 |110-1 | + 1 |-11-1-1 | + 1 |-11-10 |

Dx = - 6 [- 1 - 0] + 1[1 + 1] + 1[0 + 1] = 6 + 2 + 1 = 9

Dy = |a1c1a2c2 | = |1-610-111-1-1 |
[Find Dy.]

Dy = 1 |-11-1-1 | + 6 |011-1 | + 1 |0-11-1 |

Dy = 1 [1 + 1] + 6[0 -1] + 1[0 + 1] = 2 - 6 + 1 = - 3

Dz = |1-1-601-110-1 |
[Find Dz.]

Dz = 1 |1-10-1 | + 1 |0-11-1 | - 6 |0110 |

Dz = 1 [- 1 - 0] + 1[0 + 1] - 6[0 - 1] = - 1 + 1 + 6 = 6

x = DxD = 9 / -3 = - 3 ; y = DyD = -3 / -3 = 1;z = DzD = 6 / -3 = -2

The solution to the system is (- 3, 1, - 2).


Correct answer : (2)
 6.  
The sum of two numbers is 14 while their difference is 32. Find the numbers using Cramer's rule.
a.
25 and - 11
b.
24 and - 10
c.
26 and - 12
d.
23 and - 9


Solution:

Let the numbers be x and y.

x + y = 14

x - y = 32

AX=C
(111-1)(xy)=(1432)

[The matrix equation is AX = C.]

D = |a1b1a2b2 | = |111-1 | = 1(- 1) - 1(1) = - 2
[Find D.]

Dx = |c1b1c2b2 | = |14132- 1 | = 14(- 1) - 32(1) = - 46
[Substitute first column of A with the column of C.]

Dy = |a1c1a2c2 | = |114132 | = 1(32) - 1(14) = 18
[Substitute second column of A with the column of C.]

x = DxD = - 46- 2 = 23 and y = DyD = 18- 2 = - 9

The two numbers are 23 and - 9.


Correct answer : (4)
 7.  
William sells 3 pens and 4 pencils for $38, 5 pens and 6 pencils for $60. Find the selling price of 1 pen and 1 pencil. (Use Cramer's rule.)
a.
(5, 5)
b.
(8, 6)
c.
(6, 6)
d.
(6, 5)


Solution:

Let the selling price of 1 pen = $x

Let the selling price of 1 pencil = $y

3x + 4y = 38

5x + 6y = 60

(3456)(xy) = (3860)
[Write the matrix equation AX = C.]

D = |3456 | = 3(6) - 5(4) = - 2
[Find D.]

Dx = |384606 | = 38(6) - 60(4) = - 12
[Find Dx.]

Dy = |338560 | = 3(60) - 5(38) = - 10
[Find Dy.]

x = DxD = - 12- 2 = 6 and y = DyD = - 10- 2 = 5.
[Find x, y.]

So, the selling price of 1 pen is $6 and the selling price of 1 pencil = $5.

The solution is (6, 5).


Correct answer : (4)
 8.  
Evaluate.
|6240 |
a.
8
b.
- 8
c.
9
d.
- 9


Solution:

|6240 |

= 6(0) - 2(4) = - 8
[|abcd | = ad - bc.]


Correct answer : (2)
 9.  
Evaluate the determinant: |14- 541676 |
a.
1 3
b.
1
c.
1 2


Solution:

|14- 541676 |

= (1 / 4 × 7 / 6) - (1 / 6 × - 5 / 4)
[|abcd | = ad - bc.]

= 7 / 24 + 5 / 24 = 12 / 24 = 1 / 2


Correct answer : (3)
 10.  
If |a271 | = - 18, then find the value of a.
a.
6
b.
- 6
c.
- 4
d.
4


Solution:

|a271 | = - 18

a(1) - 2(7) = - 18
[|abcd | = ad - bc.]

a - 14 = - 18

a = - 18 + 14

a = - 4.


Correct answer : (3)

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