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

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• 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:

Page 2

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

Answered Same DayMar 16, 2022

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