Homework 5/Practice Test
Due: Wednesday, October 1
1. Consider the system of equations
x + 2y − z = 6
2x + 3z = 4
(a) Write down the augmented matrix for this system.
(b) Put the augmented matrix in reduced row echelon form (showing all steps).
(c) Write down the general form of the solution to the system.
2. Consider the matrix


1 4 0
A= 0 0 1 
2 2 6
(a) Find A−1 , if it exists.
(b) Compute det( A).
(c) Compute det( A−1 ).
3. Determine whether the following sets of vectors are linearly independent. Either way, explain your answer.
   
3 
 2



2
(a)
,
6  .


9
9

   

1
2
3


(b)  3  ,  1  ,  −1  .


−1
4
9
0
1
0
(c)
,
,
.
0
0
1
4. Let A, B ∈ Matn,n (R) and α ∈ R. Show that α ( A + B) = α A + α B.
5. Prove that a subset of a set of linearly independent vectors is linearly independent.
a b
6. (a) Suppose A =
∈ Mat2,2 (R) and that λ ∈ R. Show that det(λ A) = λ2 det( A).
c d
(b) Suppose A ∈ Matn,n (R) and λ ∈ R. Show that
det(λ A) = λ n det( A).
(H INT: Induction!)
Professor Dan Bates
Colorado State University
M369 Linear Algebra
Fall 2008
7. It turns out that the matrix

1
0
A=
5
0
0
3
0
7
2
0
6
0

0
4

0
8
has determinant det( A) = 16. Is every element of R4 in the column space of A? Explain.
Professor Dan Bates
Colorado State University
M369 Linear Algebra
Fall 2008