Inverse Matrix and Matrix Equations
A system of linear equations can be written as the in one of the following forms:
System of Linear Equations
x  2 y  4z  7

2 x  3 y  6 z  5
 3x  6 y  15 z  0

Augmented Matrix:
 1  2  4 7
 2  3  6 5


 3 6 15 0
Matrix Equation:
 1  2  4  x  7
 2  3  6   y   5 

   
 3 6 15   z  0
Looking at the last example (Matrix Equation) we can write
 1  2  4
 x
7 




C   2  3  6 and V   y  and A  5
 3 6 15 
 z 
0
then this matrix equation can be written as
CV  A
The matrix C is called the coefficient matrix
We can solve this matrix equation by the following
CV  A
C 1 (CV )  C 1 A
(C 1C )V  C 1 A
I 3V  C 1 A
Multiply both sides of the equation by C 1
Associative property
Inverse property
V  C 1 A
Identity property
(Note X, Y and Z is found by multiplying the inverse matrix of C to A)
Solving a Matrix Equation
If C is a square n  n matrix that has an inverse C 1 , and if V is a variable matrix
and A is a known matrix, both with n rows, then the solution of the matrix
equation
CV  A
is given by
V  C 1 A
Rule: To solve a system of linear equations using inverses:
1)
1)
2)
Find the inverse of the coefficient matrix
Multiply the inverse matrix to the solution matrix
a)
b)
Write the system of equations as a matrix equation.
Solve the system by solving the matrix equation.
2 x  5 y  15

3x  6 y  36
2)
Consider the following system of equations:
4 x  y  14

12 x  y  2
a)
Write the system as a matrix equation.
b)
Solve the system by solving the matrix equation.
3)
Consider the following system of equations:
5 x  3 y  4

3x  2 y  0
a)
Write the system as a matrix equation.
b)
Solve the system by solving the matrix equation.
4)
Consider the following system of equations:
3x  3 y  14

x  2 y  2
a)
Write the system as a matrix equation.
b)
Solve the system by solving the matrix equation.