1. Warm-up Problem
Use the Laplace transform to solve the IVP

2. Matrices and Eigenvalues
Ex. Let and
a) 2A =
b) A + B =

3. Ex. Let and , find AB.

4. Ex. Let and , find AB.

5. Ex.

6. AT (transpose) switches aij with aji
Ex. Let , find AT.

7. Def. If AB = I, then B is the inverse of A, written A-1.
Thm. If A is an n  n matrix, A-1 exists iff det A ≠ 0.
Thm. If , then

8. Ex. Let , find A-1.

9. Thm. Let A by n  n and let Cij = (-1)i+j Mij, where Mij is the determinant when row i and column j are removed from A. Then

10. Ex. Let , find A-1.

12. Ex. Let , find X(t) and

13. Def. Let A be n  n. A number λ is an eigenvalue of A if there is a nonzero vector K such that
AK = λK
K is called an eigenvector corresponding to λ.

14. Ex. Show that is an eigenvector of

15. Our equation was AK = λK
 AK – λK = 0
 (A – λI)K = 0
This is only true if det(A – λI) = 0
det(A – λI) = 0 is called the characteristic equation of A.

16. Ex. Find the eigenvalues and eigenvectors of

18. Ex. Find the eigenvalues and eigenvectors of

19. Ex. Find the eigenvalues and eigenvectors of