9.7 The Inverse of a Matrix

We've been learning about matrices for a while so lets learn about the inverse of a matrix.

Important: You can only have inverses of square matrices (2x2, 3x3).

Quick Review:
An identity matrix is a matrix with ones in the main diagonal and zeros for all the other elements.

Definition of an Inverse Matrix- 

Matrix A multiplied by the inverse of matrix A equals the identity of matrix A also is equal to the inverse of matrix A multiplied by matrix A.

If a square matrix has an inverse it is known as Invertible

Finding the Inverse of a Matrix

Lets find the inverse of matrix A.

The first thing we need to do is augment (or add) the multiplicative identity of A so 

Now we want to switch the left side to the identity of A by using elementary row transformations.

We have changed the left side into the identity so the right side is now the inverse

To verify the inverse plug it into

and get 

You can use this method for square matrices or just use a graphing calculator.

Solving Systems of Linear Equations

-x+3y+z=1

2x+3y=3

3x+y-2z=-2

First set up 3 matrices

Then multiply by the inverse of A

and 

so

Example from the book

Heres a link to a helpful video

http://www.youtube.com/watch?v=r9aTLTN16V4&feature=plcp

-Tommy McLeod

Thanks to geller for letting me use his account