8.9. Inverse Matrices http://www.ck12.org
8.9 Inverse Matrices
Here you will learn how to find the inverse of a matrix and how to solve a system of equations using an inverse
matrix.
Two numbers are multiplicative inverses if their product is 1. Every number besides the number 0 has a multiplica-
tive inverse. For matrices, two matrices are inverses of each other if they multiply to be the identity matrix.
What kinds of matrices do not have inverses?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/61656http://www.youtube.com/watch?v=KBYvP6YG58g James Sousa: Inverse Matrices Using Augmented Matrices
Guidance
Consider a matrixAthat has inverseA−^1. How do you find matrixA−^1 if you just have matrixA?
A=
1 2 3
1 0 1
0 2 − 1
,A−^1 =?
The answer is that you augment matrixAwith the identity matrix and row reduce.
1 2 3
1 0 1
0 2 − 1
∣∣
∣∣
∣∣
1 0 0
0 1 0
0 0 1
The row reducing is demonstrated in Example A. The right part of the augmented Matrix is the inverse matrixA−^1.
1 0 0
0 1 0
0 0 1
∣∣
∣∣
∣∣
− 113 43 13
61 −^160
3 −^13 −^13
A−^1 =
− 113 43 13
61 −^160
3 −^13 −^13
Fractions are usually unavoidable when computing inverses.
One reason why inverses are so powerful is because they allow you to solve systems of equations with the same
logic as you would solve a single linear equation. Consider the following system based on the coefficients of matrix
Afrom above.