1. The Inverse of a Square Matrix
Section 2.6
The Inverse of a Square Matrix

2. Inverse of a Matrix
Let A be a square matrix. A square matrix A-1 of equal size such that A-1A = AA-1 = I is called the inverse of A.
Ex. Given matrix A, show matrix A-1 is its inverse.
and

3. Finding the Inverse of a Matrix
Given the square matrix A.
Adjoin the identity matrix I (of the same size) to form the augmented matrix: [A | I]
Use row operations to reduce the matrix to the form: [I | B] (if possible)
Matrix B is the inverse of A.

4. Ex. Find the inverse of A.
Step 1
Step 2
Step 3

5. Step 4
Step 5
Step 6

6. Step 7
***Note: If there had been a row to the left of the vertical line in the augmented matrix containing all zeros, there would have been no inverse.

7. Formula for the Inverse of a 2 x 2 Matrix
Let
Ex.

8. Solving a System of Equations Using an Inverse
If AX = B is a linear system of equations (number of equations = number of variables) and A-1 exists, then X = A-1B is the unique solution of the system.

9. Ex. Use an inverse matrix to solve:
So (–2 , 3) is the solution
Multiply by the inverse