# Exam #2 Review

## Scheduling Information:

Exam Time: Wednesday, October 24, from 9:30-10:20AM (our usual class time).
Exam Location: HH B103 (our usual class room).
Review Session: Tuesday, October 23 from 7:00-8:30pm in WEH 8220.

## Topics Covered:

Matrix Multiplication: Section 3.1 pp. 147-151.
Matrix Inverses: Section 3.3.
Introduction to Determinants: Sectioon 4.2.
Properties of Determinants: Sectioon 4.2.
Formulas for Determinants: Sectioon 4.2.
The Eigenvalue Problem: Section 4.1.
Eigenvalues and Eigenvectors: Section 4.3.

## Review Questions:

1. What is the product of two matrices?
2. What is the Row-Column Representation of matrix multiplication?
3. What is the Matrix-Column Representation of matrix multiplication?
4. What is the Row-Matrix Representation of matrix multiplication?
5. What is the Column-Row Representation of matrix multiplication?
6. What are the properties of matrix addition and scalar multiplication like? (i.e. they are like what other properties we've studied?)
7. What important property does matrix multiplication NOT possess?
8. What is the principle of mathematical induction? How does it work?
9. What is the inverse of a matrix?
10. What is an elementary matrix?
11. What are the first three properties of the determinant? How are they different from the others?
12. What are the remaining seven properties?
13. How do elementary row operations affect the determinant?
14. How can we compute determinants by performing row operations?
15. What is meant by "expansion along a row?" "Expansion down a column?"
16. What is an eigenvalue? What is an eigenvector?
17. How can you find the eigenvalues of a matrix?
18. How can you find the eigenvectors of a matrix?
19. What vector is never an eigenvector? Why?
20. What are the eigenvalues of a diagonal matrix?
21. What do you know about the eigenvalues of an invertible matrix?
22. What can be said about the eigenvalues of a power of a matrix?
23. What are the eigenspaces of a matrix?
24. What is the span of a set of vectors?

## Exercises:

Section 3.1 #5, 9, 13, 21, 23, 27, 29, 35, 37.
Section 3.2 #19.
Section 3.3 #13, 15, 25, 27, 29, 43, 49, 53, 55.
Section 3.5 #3, 5, 7, 9.
Section 4.2 # 27, 29, 31, 33, 35, 37, 39, 45, 55.
Section 4.1 #1, 5, 7, 11.
Section 4.3 #1, 3, 5, 15, 19, 21, 23.

## Old Exam Problems:

Here are some old exam problems that I have written and given to 21-241 students in previous semesters.

Before the questions come in - no, I don't have solutions available for these problems. You have a large number of problems from the text with answers, and it's important for you to get used to working on problems where the answers are not available. You'll have a chance to ask questions at the Review Session, and during your TA's office hours, too.