Math 1104C, 1104J: Test # 1, Due Feb. 9 11:59, 2022
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• 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.
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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
answer. ∣∣∣∣∣∣
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.
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