Jonathan Greenfield Brewer MAT 275 ONLINE B Spring 2020
Assignment Section 7.3 Eigenvalues Eigenvectors due 04/25/2020 at 11:59pm MST
1. (1 point) Find the eigenvalues and eigenvectors of the ma-
trix A =−1−6
2−8
λ1=,~v1=
and
λ2=,~v2=
Correct Answers:
•<table border=’0’ cellspacing=’0’>
<tr>
<td> -4, \(\Bigl\lbrack\)</td>
<td><table border=’0’ cellspacing=’5’><tr><td> 2 </td></tr><tr><td> 1 </td></tr></table></td>
<td>\(\Bigr\rbrack\); -5, \(\Bigl\lbrack\)</td>
<td><table border=’0’ cellspacing=’5’><tr><td> -3 </td></tr><tr><td> -2 </td></tr></table></td>
<td>\(\Bigr\rbrack\)</td>
</tr>
</table>
2. (1 point) Find the eigenvalues and eigenvectors of the ma-
trix A =−2−3
2−7
λ1=,~v1=
and
λ2=,~v2=
Correct Answers:
•<table border=’0’ cellspacing=’0’>
<tr>
<td> -5, \(\Bigl\lbrack\)</td>
<td><table border=’0’ cellspacing=’5’><tr><td> 1 </td></tr><tr><td> 1 </td></tr></table></td>
<td>\(\Bigr\rbrack\); -4, \(\Bigl\lbrack\)</td>
<td><table border=’0’ cellspacing=’5’><tr><td> -3 </td></tr><tr><td> -2 </td></tr></table></td>
<td>\(\Bigr\rbrack\)</td>
</tr>
</table>
3. (1 point) Let
~v1=3
−2, ~v2=2
−3, ~v3=−5
−5.
Are the vectors~v1,~v2and ~v3linearly independent?
•choose
•linearly dependent
•linearly independent
If the vectors are independent, enter zero in every answer blank
since those are only the values that make the equation below
true. If they are dependent, find numbers, not all zero, that
make the equation below true. You should be able to explain
and justify your answer.
0
0=3
−2+2
−3+−5
−5.
Correct Answers:
•linearly dependent
•-5; 5; -1
4. (1 point) Let
~v1=
0
−15
0
, ~v2=
−6
0
−9
, ~v3=
−2
−10
−3
.
Are the vectors~v1,~v2and ~v3linearly independent?
•choose
•linearly dependent
•linearly independent
If the vectors are independent, enter zero in every answer blank
since zeros are only the values that make the equation below
true. If they are dependent, find numbers, not all zero, that
make the equation below true. You should be able to explain
and justify your answer.
0
0
0
=
0
−15
0
+
−6
0
−9
+
−2
−10
−3
.
Correct Answers:
•linearly dependent
•0.666667; 0.333333; -1
5. (1 point) Let
~v1=
−15
−4
−6
, ~v2=
10
3
4
, ~v3=
2
0
1
.
Are the vectors~v1,~v2and ~v3linearly independent?
•choose
•linearly dependent
•linearly independent
If the vectors are independent, enter zero in every answer blank
since zeros are only the values that make the equation below
true. If they are dependent, find numbers, not all zero, that
make the equation below true. You should be able to explain
and justify your answer.
1
0
0
0
=
−15
−4
−6
+
10
3
4
+
2
0
1
.
Correct Answers:
•linearly independent
•0; 0; 0
6. (1 point) Let
x(1)(t) = −et
2et,x(2)(t) = 0
et,x(3)(t) = 3et
−7et
Are the vectors x(1)(t),x(2)(t)and x(3)(t)linearly independent?
•choose
•linearly dependent
•linearly independent
If the vectors are independent, enter zero in every answer blank
since those are only the values that make the equation below
true. If they are dependent, find numbers, not all zero, that
make the equation below true. You should be able to explain
and justify your answer.
0
0=−et
2et+0
et+3et
−7et.
Correct Answers:
•linearly dependent
•-3; -1; -1
7. (1 point)
Let A=1 6
4k
Determine the value of kfor which the matrix is singular (i.e.
not invertible).
k=
Solution: The matrix is singular if det(A) = 0.
We have det(A) = 1k−24, thus the matrix is singular if k=24
1.
Correct Answers:
•24
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