1. Section 9.5 Inverses of Matrices
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2. Objectives Find the inverse of a square matrix, if it exists.
Use inverses of matrices to solve systems of equations.

3. The Identity Matrix

4. Example
For find each of the following. a) AI b) IA

5. Inverse of a Matrix
For an n  n matrix A, if there is a matrix A1 for which A1 • A = I = A • A1, then A1 is the inverse of A. Verify that is the inverse of . We show that BA = I = AB.

6. Finding an Inverse Matrix
To find an inverse, we first form an augmented matrix consisting of A on the left side and the identity matrix on the right side. Then we attempt to transform the augmented matrix to one of the form
The 2  2
identity matrix
matrix A

7. Example
Find A1, where A =

8. Example continued
Thus, A1 =

9. Notes
If a matrix has an inverse, we say that it is invertible, or nonsingular. When we cannot obtain the identity matrix on the left using the Gauss-Jordan method, then no inverse exists.

10. Solving Systems of Equations
Matrix Solutions of Systems of Equations For a system of n linear equations in n variables, AX = B, if A is an invertible matrix, then the unique solution of the system is given by X = A1B. Since matrix multiplication is not commutative in general, care must be taken to multiply on the left by A1.

11. Example
Use an inverse matrix to solve the following system of equations: 3x + 4y = 5 5x + 7y = 9 We write an equivalent matrix, AX = B: In the previous example we found A1 =

12. Example continued
We now have X = A1B. The solution of the system of equations is (1, 2).