Topic: Systems and Matrices

Subtopic: Matrix Inversion

**Overview**

To solve an algebraic equation such as 5x=12 we invert the 5 and multiply both sides by 1/5 to isolate the x. Today we do a similar process except with matrices. Matrix inversion is somewhat analogous to matrix division and will enable us to solve matrix equations. Practice working the problems algebraically and electronically. Technology comes in particularly handy when finding the inverse of a large square matrix. See the class calendar for links to my *Calculator Guide to Inverse Matrices*.

By the end of this section you should be able to solve a square consistent system of linear equations by any of the following methods:

- Algebraically by substitution or elimination methods
- Gaussian elimination to transform to row echelon form (REF) followed by back substitution
- Gauss-Jordan elimination to transform to reduced row echelon form (RREF)
- Matrix equation method (using the inverse matrix)

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 5.5.1 Check if two given matrices are inverses of one another
- 5.5.2 Recognize a multiplicative identity matrix and a multiplicative inverse matrix
- 5.5.3 Find an inverse matrix using the AI->IA
^{-1}method - 5.5.4 Use the shortcut method to finding the inverse of a 2x2 matrix (note the shortcut only works for 2x2's)
- 5.5.5 Know that if the determinant of a square matrix is zero is does not have an inverse
- 5.5.6 Solve a matrix equation AX=B by the matrix equation method (AX=B -> X=A
^{-1}B)

**Terminology**

Define:
multiplicative identity matrix (a.k.a. identity matrix), multiplicative inverse matrix (a.k.a. inverse matrix), invertible, non-invertible