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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-
을
“If
°* [
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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answered
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O a. True
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O b. False
P Flag
question
Question 4
For vectorsú and in Rºthen 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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O b. False
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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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O a. True
O b. False
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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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O a. True
O b. False
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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
А
0
0
0
B S
A B
0 A
0 0
с
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 ö onto the plane is given by
yu
ปี – นิว
projp) – +
-22-
uu น่ – น
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 Å«.
(b) Let = B.Compute the projection of v onto P.
c
A
(C) Define T:R → Ras T() = projp(6). Give the matrix of T.
[
А -AB
Let Me с с 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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