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# Math 1104C, 1104J: Test # 1, Due Feb. 9 11:59, 2022 Directives: • This document has 3 pages (including this page). • This assignment is due on March 16th at 11:59 p.m. on Brightspace. • The assignment...

Math 1104C, 1104J: Test # 1, Due Feb. 9 11:59, 2022
Directives:
• This assignment is due on March 16th at 11:59 p.m. on Brightspace.
• The assignment has 10 questions for a total of 65 points.
• You must show your work when appropriate. We look at lot more at your work than you
IMPORTANT:
or an electronic tablet or use latex.
Instructions to submit your assignment online
• You must send a scan or a picture of your work. It must clear enough so that we can read what you
wrote. We are only marking what we receive so make sure to send complete files.
• only .pdf and .jpg files are accepted.
• Make sure to submit your work at the appropriate link of the exam on Brightspace.
• Only the last version of a file or document that you submit is marked.
• No late work will be accepted.
You can contact me by email at XXXXXXXXXX.
1. (4 points) Compute the determinant of the matrix
B =
 2 −3 7−4 5 −14
8 −9 21
 ,
2. Compute the determinants of the following matrices.
(a) (4 points) ∣∣∣∣∣∣∣∣∣∣
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
∣∣∣∣∣∣∣∣∣∣
1
mailto: XXXXXXXXXX
(b) (4 points) ∣∣∣∣∣∣∣∣∣∣
XXXXXXXXXX
− XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
∣∣∣∣∣∣∣∣∣∣
3. (6 points) Let |A| =
∣∣∣∣∣∣
a b c
d e f
g h i
∣∣∣∣∣∣ = 8. Find the determinant of the following matrix. Explain you
g h i
a + 2d b + 2e c + 2f
3d 3e 3f
∣∣∣∣∣∣
4. (5 points) Consider the vectors ~v1 =
 −21
1
 , ~v2 =
 1−2
1
 , ~v3 =
 11
−2
 .
Are they linearly independent? If not
either:
(a) Find a linear dependence relation among them.
or:
(b) Express one of the vectors as a linear combination of the others.
5. (12 points) Let ~v1 =
 12
3
 , ~v2 =
 −12
1
 , ~v3 =
 1−1
1
.
(a) Show that the set B= {~v1, ~v2, ~v3} is a basis for R3.
(b) With B as in (a), find the coordinates for x =
 01
2
 in the basis B . That is, find [x]B.
(c) If [x]B =
 1−1
1
 then what is x in the standard basis?
6. (a) (4 points) Find a basis for the subspace of R4 spanned by
~u =

1
−5
3
4
 , ~v =

4
−7
2
5
 , ~w =

7
4
−9
−5
 .
(b) (1 point) Using part (a), determine whether the vectors ~u, ~v, ~w are linearly dependent.
7. (8 points) The following matrix:
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A =

1 −2 4 −1 0 5 0
−2 4 −7 1 2 −8 0
3 −6 12 − XXXXXXXXXX
2 −4 9 − XXXXXXXXXX

has reduced row echelon form: 
1 −2 0 3 0 −3 0
0 0 1 −1 0 2 0
XXXXXXXXXX
XXXXXXXXXX

Find bases for the row space of A, the column space of A and the null space of A.
8. (9 points) Conside
B =
 XXXXXXXXXX
0 0 1

(a) Find all eigenvalues of B.
(b) Find a co
esponding eigenvector for each eigenvalue in part (a).
9. (4 points) Determine which of the following vectors
~v =

1
1
1
−1
 , ~w =

1
−1
−1
2

is an eigenvector of the matrix
w =

1 1 1 1
1 −1 1 −1
1 1 −1 −1
1 −1 −1 1

and find the co
esponding eigenvalue.
10. (4 points) Let
A =
(
5 −6
3 −4
)
.
Find the eigenvalues and eigenvectors of A.
Page 3