Cramer's Rule is a method for solving systems of linear equations. It uses determinants of coefficient matrices. For a system of two equations, the solutions are determined by taking the determinants of the coefficient matrix and matrices with the constant terms replacing coefficients in the numerator. For three equations, the solutions are similarly determined using 3x3 determinants. Cramer's Rule will fail if the determinant of the coefficient matrix is zero, indicating the system is dependent with no unique solution.

1. Cramer's Rule

2. Introduction
 Cramer’s Rule is a method for solving linear
simultaneous equations. It makes use of
determinants and so a knowledge of these is
necessary before proceeding.
 Cramer’s Rule relies on determinants

3. Coefficient Matrices
 You can use determinants to solve a system of
linear equations.
 You use the coefficient matrix of the linear
system.
 Linear System Coeff Matrix
ax+by=e
cx+dy=f 





d
c
b
a

4. Cramer’s Rule for 2x2 System
 Let A be the coefficient matrix
 Linear System Coeff Matrix
ax+by=e
cx+dy=f
 If detA 0, then the system has exactly one
solution:
and
A
d
f
b
e
x
det

A
f
c
e
a
y
det


d
c
b
a
= ad – bc

5. Key Points
 The denominator consists of the coefficients
of variables (x in the first column, and y in the
second column).
 The numerator is the same as the denominator,
with the constants replacing the coefficients
of the variable for which you are solving.

6. Example - Applying Cramer’s Rule
on a System of Two Equations
Solve the system:
 8x+5y= 2
 2x-4y= -10
The coefficient matrix is: and
So:
and






 4
2
5
8
42
)
10
(
)
32
(
4
2
5
8






42
4
10
5
2




x
42
10
2
2
8



y

7. 1
42
42
42
)
50
(
8
42
4
10
5
2













x
2
42
84
42
4
80
42
10
2
2
8











y
Solution: (-1,2)

8. Applying Cramer’s Rule
on a System of Two Equations
D
D
y
D
D
x
f
c
e
a
D
d
f
b
e
D
d
c
b
a
D
f
dy
cx
e
by
ax
y
x
y
x




















14
5
3
16
3
2
y
x
y
x
19
9
10
)
3
)(
3
(
)
5
)(
2
(
5
3
3
2








D
38
42
80
)
14
)(
3
(
)
5
)(
16
(
5
14
3
16












x
D
76
48
28
)
16
)(
3
(
)
14
)(
2
(
14
3
16
2








y
D
4
19
76
2
19
38








D
D
y
D
D
x
y
x

9. Evaluating a 3x3 Determinant
(expanding along the top row)
 Expanding by Minors (little 2x2 determinants)
3
3
2
2
1
3
3
2
2
1
3
3
2
2
1
3
3
3
2
2
2
1
1
1
b
a
b
a
c
c
a
c
a
b
c
b
c
b
a
c
b
a
c
b
a
c
b
a



23
8
9
6
)
4
)(
2
(
)
3
)(
3
(
)
6
)(
1
(
2
1
0
2
)
2
(
3
1
3
2
)
3
(
3
2
3
0
)
1
(
3
2
1
3
0
2
2
3
1

















10. Using Cramer’s Rule
to Solve a System of Three Equations
Consider the following set of linear equations
11 1 12 2 13 3 1
21 1 22 2 23 3 2
31 1 32 2 33 3 3
a x a x a x b
a x a x a x b
a x a x a x b
  
  
  

11. Using Cramer’s Rule
to Solve a System of Three Equations
The system of equations above can be written in
a matrix form as:
11 12 13 1 1
21 22 23 2 2
31 32 33 3 3
a a a x b
a a a x b
a a a x b
     
     

     
     
     

12. Using Cramer’s Rule
to Solve a System of Three Equations
Define
 
   
11 12 13
21 22 23
31 32 33
1 1
2 2
3 3
a a a
A a a a
a a a
x b
x x and B b
x b
 
 
  
 
 
   
   
   
   
   
   
3
1 2
1 2 3
If 0, then the system has a unique solution
as shown below (Cramer's Rule).
, ,
D
D
D D
x x x
D D D
         
   
    

13. Using Cramer’s Rule
to Solve a System of Three Equations
where
11 12 13 1 12 13
12 22 23 1 2 22 23
13 32 33 3 32 33
11 1 13 11 12 1
2 12 2 23 3 12 22 2
13 3 33 13 32 3
a a a b a a
D a a a D b a a
a a a b a a
a b a a a b
D a b a D a a b
a b a a a b
  
  

14. Example 1
Consider the following equations:
    
 
1 2 3
1 2 3
1 2 3
2 4 5 36
3 5 7 7
5 3 8 31
where
2 4 5
3 5 7
5 3 8
x x x
x x x
x x x
A x B
A
  
   
   


 
 
 
 
 

 

15. Example 1
   
1
2
3
36
7
31
x
x x and B
x
   
   
   
   
   

   
2 4 5
3 5 7 336
5 3 8
D

   

1
36 4 5
7 5 7 672
31 3 8
D

  
 

16. Example 1
2
2 36 5
3 7 7 1008
5 31 8
D   
 
3
2 4 36
3 5 7 1344
5 3 31
D

   

1
1
2
2
3
3
672
2
336
1008
3
336
1344
4
336
D
x
D
D
x
D
D
x
D

  

   


  


17. Cramer’s Rule - 3 x 3
 Consider the 3 equation system below with variables
x, y and z:
a1x  b1y  c1z  C1
a2 x  b2 y  c2z  C2
a3x  b3y  c3z  C3

18. Cramer’s Rule - 3 x 3
 The formulae for the values of x, y and z are
shown below. Notice that all three have the
same denominator.
x 
C1 b1 c1
C2 b2 c2
C3 b3 c3
a1 b1 c1
a2 b2 c2
a3 b3 c3
y 
a1 C1 c1
a2 C2 c2
a3 C3 c3
a1 b1 c1
a2 b2 c2
a3 b3 c3
z 
a1 b1 C1
a2 b2 C2
a3 b3 C3
a1 b1 c1
a2 b2 c2
a3 b3 c3

19. Example 1
 Solve the system : 3x - 2y + z = 9
 x + 2y - 2z = -5
x + y - 4z = -2
x 
9 2 1
5 2 2
2 1 4
3 2 1
1 2 2
1 1 4

23
23
 1 y 
3 9 1
1 5 2
1 2 4
3 2 1
1 2 2
1 1 4

69
23
 3

20. Example 1
z 
3 2 9
1 2 5
1 1 2
3 2 1
1 2 2
1 1 4

0
23
 0
The solution is
(1, -3, 0)

21. Cramer’s Rule
 Not all systems have a definite solution. If the
determinant of the coefficient matrix is zero, a
solution cannot be found using Cramer’s Rule
because of division by zero.
 When the solution cannot be determined, one of
two conditions exists:
 The planes graphed by each equation are parallel
and there are no solutions.
 The three planes share one line (like three pages of
a book share the same spine) or represent the same
plane, in which case there are infinite solutions.