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Need answers to the questions in the assignment for submission.

Determine the eigenvalues and the determinant of the matrix

:-70 0

17 14 0

8 5 4

Eigenvalues from largest to smallest

Determinant

2

8

[2 -1

Determine the nullity and the rank of 3 4

2 10

0

-3

1

8

3

10

Rank

Nullity =

-2

Determine if the linear transformation T(3)

11

– 7

4

3

2 10 is onto? one-to-one? invertible?

13

0

O a. onto one-to-one: invertible

O b. not onto: one-to-one: invertible

O c.not onto; not one-to-one; not invertible

O d. onto not one-to-one: not invertible

O e not onto not one-to-one: invertible

SI

Which of the following is the inverse of

– al-

–

H

O a.

S-

Ã¬Ââ€ž

“If

Ã‚Â°* [

O d.

ols

-1S

Question 3

Let A and B ben x n matrices and det(AB)

= 1, then both A and B are invertible.

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O a. True

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Question 4

For vectorsÃƒÂº and in RÃ‚Âºthen proj:) is perpendicular to i.

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O a. True

O b. False

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Question 5

Any real polynomial of degree 2 has exactly 2 real roots.

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Question 6

Let A be a 5 x 6 matrix with nullity 3. Then the rank of A is 2.

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Question 7

Let T be a linear transformation defined by T(T) = Az, where A is a 4 x 3 matrix. Then I cannot be onto.

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Question 8

The characteristic polynomial of a 4 x 4 matrix always have four real solutions.

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Question

Let L, be the line passing through the points Q1 = (A,B,C) and Q2 = (A+1, B-1,C).

Let Ly be the line passing through the point P = (-A, -B-D) with direction vector

k

Find the value of k so that the two lines intersect, and determine the point of intersection.

Find the condition on s such that the eigenspace of

ÃÂ

0

0

0

B S

A B

0 A

0 0

Ã‘Â

D

B

A

associated with the eigenvalue 1 = A has dimension two.

Let P be a plane containing the origin in R. Let and be two non-zero vectors that are orthogonal to each other. For any vector ERP the projection of ÃƒÂ¶ onto the plane is given by

yu

Ã Â¸â€ºÃ Â¸Âµ – Ã Â¸â„¢Ã Â¸Â´Ã Â¸Â§

projp) – +

-22-

uu Ã Â¸â„¢Ã Â¹Ë† – Ã Â¸â„¢

D

Let P be the plane +Dy + z = 0 and let i

[1]

0

(a) Find a non-zero vector iz in the plane that is orthogonal to Ã…Â«.

(b) Let = B.Compute the projection of v onto P.

c

A

(C) Define T:R Ã¢â€ â€™ Ras T() = projp(6). Give the matrix of T.

[

ÃÂ -AB

Let Me Ã‘Â Ã‘Â D

0 0 0 1

(a) Determine the characteristic polynomial of M.

(b) Determine the eigenvalues of Mand their algebraic multiplicities.

() For each eigenvalue of M. determine the basic eigenvector(s).

(d) Give an invertible matrix P and a diagonal matrix D such that M=PDP-1

Suppose that the sequence 2o, 11, 12, 13,… is defined by the initial conditions 2o = A 2 = B. and the recurrence relation

#2=

(C+2)Ik+1-(C+1).Ik

for ko.

(a) Find a 2 x 2 matrix M satisfying

IR

J

=M

Io

for k >0.

161

(b) Give an invertible matrix P and a diagonal matrix D such that M = PDP-1

(c) Use the diagonalization of M to find a general formula of Ik

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