Cramer's Rule

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

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

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

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.

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

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)

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

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

















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
  
  
  

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
     
     

     
     
     

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
         
   
    

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
  
  

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
  
   
   


 
 
 
 
 

 

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

  
 

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

  

   


  


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

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

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

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)

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.