Notes, Lesson 6.4
Identity and Inverse Matrices
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We have already learned about the identity element for multiplication. There also are identity elements for matrices under multiplication. They are called identity matrices.


Identity Matrix The identity matrix I for multiplication is a square matrix with a 1 for every element of the principal diagonal (top left to bottom right) and a 0 in all other positions.
The identity matrix for   is   because 

Back in multiplication, you know that 1 is the identity element for multiplication. Whenever the identity element for an operation is the answer to a problem, then the two items operated on to get that answer are inverses of each other.

This is also true in matrices. If an identity matrix is the answer to a problem under matrix multiplication, then each of the two matrices is an inverse matrix of the other.


Inverse Matrix The matrix which when multiplied by the original matrix gives the identity matrix as the solution.
When solving equations like 8x=72, you can use the ERAA and multiply both sides of the equation by the multiplicative inverse of 8, to get x=9.

Likewise we will (in the next lesson) use an inverse matrix to multiply both sides of a matrix equation to solve the equation.

To find an inverse matrix:

Sample Problem.
Find the inverse of: 
Given Problem.
First check if the determinant is zero. If it is not zero, then the inverse we are trying to find exists.
Now use the inverse formula to find and calculate the inverse matrix.
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