Homework 2
Due: Wednesday, September 10
1. Write the system of linear equations
2x − 3y + z = 8
x − 2y − 2z = 4
in the following three formats:
(a) as an augmented matrix,
(b) in matrix-vector format (i.e., Ax = b), and
(c) as a linear combination of the three columns of A.
2. Suppose ~v ∈ Rn is a solution to the homogeneous matrix equation
Ax = 0.
Suppose λ ∈ R. Show that λ~v is also a solution to this equation.
3. Prove part (2) of Theorem MMDAA on page 211 of Beezer: If B, C ∈ Mat m,n (R) and D ∈
Mat n,p (R), then ( B + C ) D = BD + CD. (H INT: Beezer proves part (1), and uses the notation
Rm×n for Matm,n (R).)
4. Our three elementary row operations can be captured as multiplication (on the left) by special elementary matrices. For example, for a 3 × 3 matrix, you can swap rows 1 and 3 by
multiplying (on the left) by


0 0 1
 0 1 0 .
1 0 0
For example:

 
 
7 8 9
1 2 3
0 0 1
 0 1 0 · 4 5 6  =  4 5 6 
1 2 3
7 8 9
1 0 0

(a) What is the elementary matrix corresponding to multiplying row 2 of a 3 × 3 matrix by
α?
(b) What is the elementary matrix corresponding to adding β times row 3 to row 1 in a
3 × 3 matrix?
5. Consider the matrices A and B:
A=
Professor Dan Bates
Colorado State University
2 4
3 1
B=
1
− 10
3
10
2
5
− 15
M369 Linear Algebra
Fall 2008
(a) Demonstrate that A and B are inverses.
(b) Solve the following system of linear equations by writing it in matrix-vector form and
using B appropriately:
2x + 4y = 3
3x + y = 1
(Notice that you can check your answer using A.)
6. For each of the following matrices, does the matrix have an inverse? If so, find it. If not, why
not?


1 0 3
(a)  2 1 8 
1 2 7


2 0 4
(b)  0 6 0 
6 0 8
Professor Dan Bates
Colorado State University
M369 Linear Algebra
Fall 2008