USING THE INVERSE MATRIX METHOD
to solve systems of equations
The concept of solving systems of equations using inverse matrices is similar to the
concept of solving simple equations.
For example, to solve 7 x  14 , we multiply both sides by the multiplicative inverse
of 7, which is 1/7:
1
1
 7 x   14
7
7
On the left, (1/7) × 7 = 1. The number 1 is the “identity” for multiplication of
ordinary numbers.
The solution for our equation is:
x2
We will extend this concept to solving systems of simultaneous equations.
In general, if we want to solve a system of equations:
Let
Then
ax  by  e
cx  dy  f
a b
A

c d 
 x
X  
 y
e
B 
f
Coefficient
Matrix
Variable
Matrix
Product
Matrix
AX  B
If we multiply each side by A-1, we have:
However, we know that A1 A  I , the identity matrix. So we obtain:
But, IX  X , so the solution to the system of equations is given by:
Example: Solve
2 x  11y  25
3x  2 y  6
Solution:
Write the system in the form of a matrix equation AX  B .
Multiply both sides of the matrix equation by A–1 and solve.