# 21-241 Matrix Algebra

# Homework

### Homework #1:

**Exercises:** Due Wednesday, Sept 3

Section 1.1: 2, 8, 14, 15, 20, 25, 33, 34
Section 1.2: 3, 7, 12, 14, 16, 23, 29, 31
Section 1.3: 3, 7, 10, 12, 13, 17, 22

### Homework #2:

**Exercises:** Due Wednesday, Sept 10

Section 1.4: 10, 12, 15, 19, 35
Section 1.5: 2, 6, 16, 29-32, 35
Section 1.6: 4, 6, 12

### Homework #3:

**Exercises:** Due Wednesday, Sept 17

Section 1.7: 3, 6, 12, 20, 23, 28, 31, 33
Section 1.8: 4, 10, 12, 15, 17, 18, 25, 31
Section 1.9: 3, 6, 7, 9, 13, 17, 25, 29-31

### Exam #1:

In class, Monday, Sept 22
Review Sheet

### Homework #4:

**Exercises:** Due Monday, Sept 29

Section 2.1: 9, 10, 13, 14, 23, 24, 27, 28
Section 2.2: 2, 7, 13-16, 20, 31, 33

### Homework #5:

**Exercises:** Due Wednesday, October 8

Section 2.3: 7, 14, 18, 19, 22, 28, 33
Section 2.7: 2, 5, 7, 14, 17, 18
Problem A: Refer to the triangle in Exercise 2.7.2. Suppose you wish
to rotate this triangle 60 degrees with respect to the point (6,8).
Draw a sketch which shows the initial position of the triangle, and illustrates approximately where the triangle would
end up after this transformation. (You do not need to try and guess the
coordinates of the new vertices -- just illustrate their approximate
positions.) Then use the matrix you found in Exercise 2.7.7 to perform
this rotation. This will give you the new vertices. Finally,
draw another sketch which illustrates the transformation.
Section 2.8: 3,6,11-15,24,26,27

### Homework #6:

**Exercises:** Due Wednesday, Oct 15

Section 2.9: 1, 5, 12, 14, 16, 20
Section 3.1: 7, 11, 23, 37, 38
Section 3.2: 1, 8, 13, 23, 26, 31, 34-36

### Homework #7:

**Exercises:** Due Wednesday, Oct 22

Section 5.1: 6, 10, 16-20, 27, 29, 30, 33
Section 5.2: 1, 8, 10, 17, 18, 20

### Exam #2:

In class, Friday, Oct 24

Review Sheet

Review Sheet Solutions

### Homework #8:

**Exercises:** Due Wednesday, Nov 5

Section 5.3: 1, 3, 15, 18, 24, 27, 31
Section 5.5: 5
Problem A: Find all real and complex eigenvalues of the 3 X 3
matrix A below, and for each eigenvalue, find a corresponding
eigenvector.
[1 1 -1]
A = [0 1 0]
[1 0 1]

Problem B: Find all real and complex roots of the polynomial
x^3 + x^2 + 17x - 87

Hints: First look for real integer roots by substituting values
such as 0, 1, -1, 2, -2, etc., into the polynomial, in hopes that you
will find a
"nice" root r. If you find one, then you know that the polynomial can
be factored as follows:
(x - r) (x^2 + ax + b)

By multiplying these factors, you can determine the appropriate values
for
a and b. Finally, you can find the roots of the quadratic factor,
using the
quadratic formula if necessary.
Problem C: Find all real and complex eigenvalues of the 4 X 4
matrix A below, and for each eigenvalue, find a corresponding
eigenvector.
[1 -1 1 -1]
A = [1 1 1 1]
[0 0 1 1]
[0 0 1 1]

Section 5.6: 3, 10, 12, 14, 15, 17ab

### Homework #9:

**Exercises:** Due Wednesday, Nov 12

Section 6.1: 2, 3, 6, 8, 9, 11, 14-18, 28
Section 6.2: 2, 3, 7, 10, 12, 13, 17, 20, 22, 25, 28

### Homework #10:

**Exercises:** Due Wednesday, Nov 19

Section 6.3: 1, 5, 9, 12, 13, 16, 17
Section 6.4: 3, 7, 11, 12
Section 6.5: 19, 20, 22
Problem A: The hypercube is the set of points in R^4 with 0 <=
xi <=1 for i=1,2,3,4. Find the extreme points ("corners") when the
hypercube is sliced by the following planes:

(a) x1+x2+x3+x4 = 1

(b) x1+x2+x3+x4 = 1.5

What (3-dimensional) solids are these?

### Exam #3:

In class, Monday, Nov 24

Review Sheet

Review Sheet Solutions

### Homework #11:

**Exercises:** Due NEVER

Section 7.1: 1-12, 16, 18, 21, 24, 27-30
Section 7.2: 1, 5, 8, 12, 19
Section 7.3: 5, 8, 10, 11

### Final Exam

Baker Hall 136A, 1-4 PM, Thursday, Dec 11

Review Sheet

Review Sheet Solutions