Solving using systems

Using matrices with systems
You can use matrices to solve systems of
equations. This is most useful for large numbers
of equations, though we will only use 2 or 3
variables.

Example
1 1 3
2 1 1
x
y
     
           
AX = B
3
2 1
x y
x y
 
  
Matrix of
coefficients
Matrix of variables
Matrix of
sum or
difference
As all matrices:

Example
Find the inverse to cancel the coefficient matrix.
1 1
(1)( 1) (2)( 1) 1
2 1

    

Inverse:
1 1 1 1 11
2 1 2 11
A     
        

Using with systems
1 1 1 1 1 1 3
2 1 2 1 2 1 1
x
y
           
                     
1 1
AX B
A AX A B 


A matrix times its own
inverse equals the identity
matrix, so it cancels.
The inverse must go
FIRST on both sides.
(Multiplication order
matters!)

Using with systems
(-4,-7)
1 1 3
2 1 1
x
y
     
           
4
7
x
y
   
      

Cramer’s Rule
Cramer’s Rule also allows you to solve systems
of equations using matrices. A matrix is set up
of just the coefficients, then one more matrix for
each variable.

Example
First, write the coefficients as a matrix.
4 3
6 2
C
 
   

Example, continued
Make a matrix for each variable by replacing a
column with the answers to each equation.
Use these numbers to
replace the x or y column.
6 3
11 2
xC
  
   
4 6
6 11
yC
 
    

Example, continued
Find all the determinants
4 3
6 2
C
 
   
6 3
11 2
xC
  
   
4 6
6 11
yC
 
    
Determinant C:
(4)(2) – (-6)(-3)
= -10
Determinant Cx:
(-6)(2) – (-11)(-3)
= -45
Determinant Cy :
(4)(-11) – (-6)(-6)
= -80

Example
Calculate values
det
det
45
10
4.5
xC
x
C
x
x





det
det
80
10
8
yC
y
C
y
y






Solving using systems

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