1. Ch. 7 – Matrices and Systems of Equations
7.6 – Inverses of Square Matrices

2. Inverses
Identity Matrix (I)= a square matrix with 1s in the main diagonal and 0s everywhere else
Two matrices are inverses of each other if they multiply to the identity matrix
The inverse of A is denoted A-1
A is said to be invertible if it has an inverse
Ex: Prove that B is the inverse of A.
Multiply A and B and see if you get the identity matrix! 1 1

3. Inverses Ex 1: Find the inverse of A.
We want to solve A A-1 = I for A-1, but how?
Matrix multiplication tells us that we want to solve the following two augmented matrices simultaneously:
…so we do that by reducing the adjoined matrix [ A : I ] !
When finding inverses of square matrices, RREF the augmented matrix !
In this problem, we need to RREF the following matrix:

4. Inverses Ex 1 (cont’d): Find the inverse of A.
Add E2 and E1 and replace for E2…
Add 6(E2) and E1 and replace for E1…

5. Inverses Ex 1 (cont’d): Find the inverse of A.
So the inverse of A is the right side of the RREF’d matrix.
Divide E1 by 2…

6. Ex 2 : Find the inverse of A.
RREF the adjoined matrix!
Add -(E2) and E1 and replace for E2…
Add -6(E1) and E3 and replace for E3…

7. Ex 2 (cont’d): RREF the adjoined matrix!
Add -(E1) and E2 and replace for E1…
Add 4(E2) and E3 and replace for E3…

8. Ex 2 (cont’d): RREF the adjoined matrix!
Add -(E1) and E3 and replace for E1…
Add -(E2) and E3 and replace for E2…

9. Check your solution with your calculator!
Ex 2 (cont’d):
Check your solution with your calculator!
Separate the inverse matrix from the identity to get…

10. An alternative method to solving systems
of equations involves using inverse functions.
Ex 3: Use an inverse matrix to solve the system.
If we write the system in AX = B form, where A is the coefficient matrix, X is the variable matrix, and B is the constant term matrix, it will look like this:
By multiplying A-1 in front of both sides of the equation, we get A to cancel and an equation in X = A-1B form: