3.8
Use
Inverse Matrices to Solve Linear Systems
Identity matrix
An n ´ n matrix with 1's on the main diagonal and 0's elsewhere
Inverse matrices
Two n ´ n matrices A and B are
inverses of each other if their product (in both orders) is the n ´ n identity matrix.
Matrix of variables
The matrix of variables of the linear system .
Matrix of constants
The matrix of constants of the linear system .
THE INVERSE OF A 2 ´ 2 MATRIX
ad -
cb | A |
The inverse of the
matrix A =
provided _ad - cd_ ¹ 0.
Example 1
Find the inverse of a2 ´ 2 matrix
Find the inverse of A =
-12 + 8
A-1 =
You can check
the inverse by showing that AA-1 = I = A-1A.
=
Example
2
Solve
a matrix equation
Solve
the matrix equation AX = B for the 2 ´ 2
matrix X.
Begin
by finding the inverse of A.
A-1 =
To solve the equation for X, multiply both sides of the equation by A-1 on the left.
= A-1AX
= A-1B
IX
= A-1B
X =
A-1B
Example
3
Find
the inverse of a 3 ´ 3 matrix
Use
a graphing calculator to find the inverse of A. Then use the calculator
to verify your result.
A =
Solution
Enter the matrix A into a graphing calculator and calculate AA-1.Then compute _A-1A_ _A-1A_ to verify that you obtain the _3 ´ 3_ identity matrix.
USING AN INVERSE MATRIX TO
SOLVE A LINEAR SYSTEM
Step 1 Write the system as a matrix equation AX
= B. The matrix A is the _coefficient_
matrix, X is the matrix of _variables_, and B is the matrix of _constants_.
Step 2 Find the inverse of matrix A.
Step 3 Multiply
each side of AX = B by A-1 on the _left_ to find the solution X = A-1B.
Example 4
Solve a linear system
Use an inverse
matrix to solve the linear system.
2x + 3y = 15 Equation
1 x - 2y = -17 Equation 2
Solution
1. 
Write the linear system as a matrix equation AX = B.
2.
If A does
not have an inverse, then the system has either no solution or infinitely
many solutions.
Find the inverse of matrix A.
3. 
Multiply the matrix of constants by A-1 on the left.
The solution of the system is (_-3_,_7_).