MAT 1234 Calculus I

1. MAT 1234Calculus I 9.6 Inverses of Matrices http://myhome.spu.edu/lauw

2. HW • WebAssign 9.6 • Wednesday/Monday Quiz: 9.4, 9.5, 9.6

3. Recall (9.4) • Introduce Elementary Row Operations • Gauss-Jordan Eliminations

4. Recall (9.5)

5. 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. • A second method of solving system of linear equations.

6. Recall Identity Matrix nxn Square Matrix

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-1A

8. 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-1A • If such inverse A-1 exists, then the matrix A is said to be invertible.

9. 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-1A • If such inverse A-1 exists, then the matrix A is said to be invertible. • We will focus only on invertible matrices (to save time).

10. Comparison

11. Example 1 (a)

12. How to Find Inverses? • 2x2 -> Formula • nxn -> Row Operations

13. Inverse of a 2x2 Matrix

14. Example 1 (b)

15. Matrix Equations

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

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. Idea The row operations “effectively” multiply the matrix A by A-1to get I.

20. Example 2 (a) Find the inverse of

21. Example 2 (a)

22. Example 2 (b) (=9.4 Example 2)

23. Example 2 (b) (=9.4 Example 2)

24. 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: