1. MAT 2401 Linear Algebra 2.3 The Inverse of a Matrix http://myhome.spu.edu/lauw

2. HW WebAssign 2.3 Written HW

3. Preview The definition of the Inverse of a matrix. Formula of the inverse for 2x2 matrices. Use row operations to find the inverse of nxn matrices.

4. Recall

5. Recall Identity Matrix nxn Square Matrix

6. Inverse Matrix Let A be a square matrix. Then the inverse for A is a square matrix A -1 of the same size as A such that AA -1 = I = A -1 A

7. Inverse Matrix Let A be a square matrix. Then the inverse for A is a square matrix A -1 of the same size as A such that AA -1 = I = A -1 A If such inverse A -1 exists, then the matrix A is said to be invertible (otherwise, singular).

8. Example 1 (a)

9. Inverse of a 2x2 Matrix

10. Example 1 (b)

11. Matrix Equations

12. Example 1 (c) Use matrix inverse to solve

13. Example 1 (c) Use matrix inverse to solve

14. Remarks When n≥3, there are no useful formula to find the inverse of an invertible matrix.

15. How to Find A -1 ?

16. Example 1 (d) Use row operations to find the inverse of

17. Inverse of an 3x3 Matrix (Same for nxn matrices) Given matrix A, we set up the following matrix

18. Inverse of an 3x3 Matrix (Same for nxn matrices) Given matrix A, Use row operations to get to the second matrix. A -1 (if exists) is the matrix on the right half.

19. Example 2 (a) Find the inverse of

20. Example 2 (b)

22. Q&A We can use two methods to solve a system of equations. (a) Gauss-Jordan Elimination (b) Matrix Inverse Q: Why use (b) when (a) is easier? A:

23. Properties of Matrix Inverses If A be an invertible matrix, k  Z +, c≠0 is a scalar, then A -1, A k, cA, and A T are invertible. Also, 1. (A -1 ) -1 = A 2. (A k ) -1 =(A -1 ) k 3. (cA) -1 = A -1 4. (A T ) -1 =(A -1 ) T 5. (AB) -1 =B -1 A -1

24. Cancellation Properties If C is invertible, then 1. AC=BC implies A=B 2. CA=CB implies A=B