## MATH1014 Quizzes

Quiz 10: Introduction to Eigenvalues and Eigenvectors
Question 1 Questions
Which of the following are eigenvectors of $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]$? (More than one answer may be correct.) (Zero or more options can be correct)
 a) $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$ b) $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$ c) $\left[\begin{array}{c}\hfill 5\hfill \\ \hfill 5\hfill \end{array}\right]$ d) $\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 6\hfill \end{array}\right]$ e) $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -3\hfill \end{array}\right]$ f) $\left[\begin{array}{c}\hfill -3\hfill \\ \hfill 1\hfill \end{array}\right]$

There is at least one mistake.
For example, choice (a) should be True.
$\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]=3\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$
There is at least one mistake.
For example, choice (b) should be False.
$\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 3\hfill \end{array}\right]$, which is not a multiple of $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$.
There is at least one mistake.
For example, choice (c) should be True.
$\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill 5\hfill \\ \hfill 5\hfill \end{array}\right]=3\left[\begin{array}{c}\hfill 5\hfill \\ \hfill 5\hfill \end{array}\right]$
There is at least one mistake.
For example, choice (d) should be True.
$\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 6\hfill \end{array}\right]=-\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 6\hfill \end{array}\right]$
There is at least one mistake.
For example, choice (e) should be True.
$\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -3\hfill \end{array}\right]=-\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -3\hfill \end{array}\right]$
There is at least one mistake.
For example, choice (f) should be False.
$\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill -3\hfill \\ \hfill 1\hfill \end{array}\right]=\left[\begin{array}{c}\hfill -5\hfill \\ \hfill -9\hfill \end{array}\right]$, which is not a multiple of $\left[\begin{array}{c}\hfill -3\hfill \\ \hfill 1\hfill \end{array}\right]$.
Correct!
1. True $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]=3\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$
2. False $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 3\hfill \end{array}\right]$, which is not a multiple of $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$.
3. True $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill 5\hfill \\ \hfill 5\hfill \end{array}\right]=3\left[\begin{array}{c}\hfill 5\hfill \\ \hfill 5\hfill \end{array}\right]$
4. True $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 6\hfill \end{array}\right]=-\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 6\hfill \end{array}\right]$
5. True $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -3\hfill \end{array}\right]=-\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -3\hfill \end{array}\right]$
6. False $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill -3\hfill \\ \hfill 1\hfill \end{array}\right]=\left[\begin{array}{c}\hfill -5\hfill \\ \hfill -9\hfill \end{array}\right]$, which is not a multiple of $\left[\begin{array}{c}\hfill -3\hfill \\ \hfill 1\hfill \end{array}\right]$.
Which one of the following is an eigenvector of $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill -4\hfill & \hfill 5\hfill \end{array}\right]$? Exactly one option must be correct)
 a) $\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 1\hfill \\ \hfill 5\hfill \end{array}\right]$ b) $\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 1\hfill \\ \hfill 4\hfill \end{array}\right]$ c) $\left[\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \\ \hfill 4\hfill \end{array}\right]$

Choice (a) is incorrect
$A\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 1\hfill \\ \hfill 5\hfill \end{array}\right]=\left[\begin{array}{c}\hfill -6\hfill \\ \hfill 3\hfill \\ \hfill 13\hfill \end{array}\right]$, which is not a multiple of $\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 1\hfill \\ \hfill 5\hfill \end{array}\right]$.
Choice (b) is correct!
$A\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 1\hfill \\ \hfill 4\hfill \end{array}\right]=2\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 1\hfill \\ \hfill 4\hfill \end{array}\right]$
Choice (c) is incorrect
$A\left[\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \\ \hfill 4\hfill \end{array}\right]=\left[\begin{array}{c}\hfill -4\hfill \\ \hfill 6\hfill \\ \hfill 32\hfill \end{array}\right]$, which is not a multiple of $\left[\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \\ \hfill 4\hfill \end{array}\right]$.
Given that $\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 7\hfill \end{array}\right]$ is an eigenvector of $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -3\hfill \end{array}\right]$, what is the corresponding eigenvalue? (Enter your answer into the answer box.)

Correct!

The eigenvalue is $\lambda$ such that $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -3\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 7\hfill \end{array}\right]=\lambda \left[\begin{array}{c}\hfill 0\hfill \\ \hfill 7\hfill \end{array}\right]$.
Given that $\left[\begin{array}{c}\hfill 6\hfill \\ \hfill 6\hfill \end{array}\right]$ is an eigenvector of $\left[\begin{array}{cc}\hfill 3\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill 0\hfill \end{array}\right]$, what is the corresponding eigenvalue? (Enter your answer into the answer box.)

Correct!

The eigenvalue is $\lambda$ such that $\left[\begin{array}{cc}\hfill 3\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill 6\hfill \\ \hfill 6\hfill \end{array}\right]=\lambda \left[\begin{array}{c}\hfill 6\hfill \\ \hfill 6\hfill \end{array}\right]$.
Given that $\left[\begin{array}{c}\hfill -3\hfill \\ \hfill 3\hfill \\ \hfill 12\hfill \end{array}\right]$ is an eigenvector of $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill -4\hfill & \hfill 5\hfill \end{array}\right]$, what is the corresponding eigenvalue? (Enter your answer into the answer box.)

Correct!

The eigenvalue is $\lambda$ such that $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill -4\hfill & \hfill 5\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill -3\hfill \\ \hfill 3\hfill \\ \hfill 12\hfill \end{array}\right]=\lambda \left[\begin{array}{c}\hfill -3\hfill \\ \hfill 3\hfill \\ \hfill 12\hfill \end{array}\right]$.
The solutions to which one of the following equations are the eigenvalues of $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 3\hfill \\ \hfill -2\hfill & \hfill 8\hfill \end{array}\right]$? Exactly one option must be correct)
 a) ${\lambda }^{2}-10\lambda +22=0$ b) ${\lambda }^{2}-10\lambda +10=0$ c) ${\lambda }^{2}-6\lambda +22=0$ d) ${\lambda }^{2}-6\lambda +22=0$

Choice (a) is correct!
Choice (b) is incorrect
Try again. The equation is $\left|\begin{array}{cc}\hfill 2-\lambda \hfill & \hfill 3\hfill \\ \hfill -2\hfill & \hfill 8-\lambda \hfill \end{array}\right|=0$.
Choice (c) is incorrect
Try again. The equation is $\left|\begin{array}{cc}\hfill 2-\lambda \hfill & \hfill 3\hfill \\ \hfill -2\hfill & \hfill 8-\lambda \hfill \end{array}\right|=0$.
Choice (d) is incorrect
Try again. The equation is $\left|\begin{array}{cc}\hfill 2-\lambda \hfill & \hfill 3\hfill \\ \hfill -2\hfill & \hfill 8-\lambda \hfill \end{array}\right|=0$.
What are the eigenvalues of the matrix $\left[\begin{array}{cc}\hfill -3\hfill & \hfill 5\hfill \\ \hfill 0\hfill & \hfill 8\hfill \end{array}\right]$? Exactly one option must be correct)
 a) $-3,5$ b) $-3,0$ c) $-3,8$ d) $5,8$ e) $5,0$

Choice (a) is incorrect
Since the matrix is triangular, the eigenvalues are the diagonal entries.
Choice (b) is incorrect
Since the matrix is triangular, the eigenvalues are the diagonal entries.
Choice (c) is correct!
Choice (d) is incorrect
Since the matrix is triangular, the eigenvalues are the diagonal entries.
Choice (e) is incorrect
Since the matrix is triangular, the eigenvalues are the diagonal entries.
Find the eigenvalues of the matrix $\left[\begin{array}{cc}\hfill -7\hfill & \hfill 3\hfill \\ \hfill -6\hfill & \hfill 4\hfill \end{array}\right]$. Exactly one option must be correct)
 a) $1,10$ b) $2,-5$ c) $-7,4$ d) $-2,5$

Choice (a) is incorrect
Try again. Find $\lambda$ such that $\left|\begin{array}{cc}\hfill -7-\lambda \hfill & \hfill 3\hfill \\ \hfill -6\hfill & \hfill 4-\lambda \hfill \end{array}\right|=0.$
Choice (b) is correct!
Choice (c) is incorrect
Try again. Find $\lambda$ such that $\left|\begin{array}{cc}\hfill -7-\lambda \hfill & \hfill 3\hfill \\ \hfill -6\hfill & \hfill 4-\lambda \hfill \end{array}\right|=0.$
Choice (d) is incorrect
Try again. Find $\lambda$ such that $\left|\begin{array}{cc}\hfill -7-\lambda \hfill & \hfill 3\hfill \\ \hfill -6\hfill & \hfill 4-\lambda \hfill \end{array}\right|=0.$
Given that $0$ is an eigenvalue of $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right]$ find all the corresponding eigenvectors. Exactly one option must be correct)
 a) $\left[\begin{array}{c}\hfill 2t\hfill \\ \hfill t\hfill \end{array}\right],t\in ℝ,t\ne 0.$ b) $\left[\begin{array}{c}\hfill t\hfill \\ \hfill 2t\hfill \end{array}\right],t\in ℝ,t\ne 0.$ c) $\left[\begin{array}{c}\hfill -t\hfill \\ \hfill 2t\hfill \end{array}\right],t\in ℝ,t\ne 0.$ d) $\left[\begin{array}{c}\hfill -2t\hfill \\ \hfill t\hfill \end{array}\right],t\in ℝ,t\ne 0.$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Given that $-3$ is an eigenvalue of $\left[\begin{array}{cc}\hfill 5\hfill & \hfill -1\hfill \\ \hfill 8\hfill & \hfill -4\hfill \end{array}\right]$ find all the corresponding eigenvectors. Exactly one option must be correct)
 a) $\left[\begin{array}{c}\hfill t\hfill \\ \hfill t\hfill \end{array}\right],t\in ℝ,t\ne 0.$ b) $\left[\begin{array}{c}\hfill t\hfill \\ \hfill -t\hfill \end{array}\right],t\in ℝ,t\ne 0.$ c) $\left[\begin{array}{c}\hfill t\hfill \\ \hfill 8t\hfill \end{array}\right],t\in ℝ,t\ne 0.$ d) $\left[\begin{array}{c}\hfill 8t\hfill \\ \hfill t\hfill \end{array}\right],t\in ℝ,t\ne 0.$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect