2. RECORD

4. Gaussian Elimination:
derived system
back-substitution

5. Exercise: Solve the system by Gaussian elimination:

7. The Inverse of a Matrix:
A is the inverse of B
B is the inverse of A
Example:

8. Example: Matrices with zero rows or columns have no inverses
properties of the matrix
inverse

9. An inverse of a product of two and more invertible matrices
Example: verify
Generalize for a product of any number of invertible matrices:

10. Exercise:
A square matrix A satisfies A² -3A+I = 0. Find the inverse of A.

11. Invertability of a 2 x 2 Matrix:
Exercise: Verify.
Exercise: Find A.

12. Example: an inverse of a diagonal matrix

13. Elementary Matrices:
Examples: Inspect the effect of multiplication involving “special” matrices.

14. Elementary Matrices: A simple Way to Construct an Elementary Matrix
Examples:

15. Exercise:

16. Inverses of Elementary Matrices:

17. Exercise: Find inverses for elementary matrices

18. Invertability of a Square Matrix:

19. Computing the Inverse:
convert to the identity matrix
by row operations

20. Example: Find the inverse of A.

21. Why are we interested in finding the matrix inverse?
Solving Matrix equations with Inverses:
Example: Solve the system using the inverse.

22. LU-decomposition
“A has an
LU-decomposition”
Theorem
A allows an
LU-decomposition
Theorem

23. Solving SLE equations using LU-decomposition:

24. Corresponding
elementary
matrices

25. lower-∆
A =
L U

26. Exercise: find an LU-decomposition