1. Inverse of a Square Matrix
Section 2.6
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.
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. Step 1
Step 2
Step 3
. . .

5. Step 4
Step 5
Step 6

6. ***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. Step 1
Step 2
. . .

8. Step 3
Since all entries in the last row of the submatrix
that comprises the left-hand side of the augmented
matrix just obtained are all equal to zero, the latter
cannot be reduced to the form [ I | B ].
Accordingly, we draw the conclusion that A is
singular – that is, does not have an inverse.

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

10. 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.

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

12. Homework #1 1-6 all, 9 – 19odd , 21 – 24 all, 31- 34 all
#2 P – 6 all, 7 – 23 eoo
#3 P odd
#4 P122 5, 9, 13