MATH1014 Quizzes

Quiz 12: Eigenvalues and Eigenvectors
Question 1 Questions
Let $A=\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -2\hfill & \hfill 3\hfill \end{array}\right]$, ${v}_{1}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}\right]$, ${v}_{2}=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 2\hfill \end{array}\right]$, ${v}_{3}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill -1\hfill \end{array}\right]$and ${v}_{4}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 4\hfill \end{array}\right]$.
Which of the following statements is correct ? Exactly one option must be correct)
 a) ${v}_{1}$ and ${v}_{2}$ are eigenvectors of $A$ b) ${v}_{1}$ and ${v}_{3}$ are eigenvectors of $A$ c) ${v}_{2}$ and ${v}_{3}$ are eigenvectors of $A$ d) ${v}_{3}$ and ${v}_{4}$ are eigenvectors of $A$

Choice (a) is correct!
Choice (b) is incorrect
$A{v}_{3}$ is not a multiple of ${v}_{3}$.
Choice (c) is incorrect
$A{v}_{3}$ is not a multiple of ${v}_{3}$.
Choice (d) is incorrect
$A{v}_{3}$ is not a multiple of ${v}_{3}$; $A{v}_{4}$ is not a multiple of ${v}_{4}$.
$\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill 2\hfill \end{array}\right]$ is an eigenvector of $\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -2\hfill & \hfill 3\hfill \end{array}\right]$. What is the corresponding eigenvalue? (Enter your answer into the answer box.)

Correct!

Incorrect. Please try again.
Multiply $\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -2\hfill & \hfill 3\hfill \end{array}\right]$ by $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill 2\hfill \end{array}\right]$. The result should be a multiple of $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill 2\hfill \end{array}\right]$.
$\left[\begin{array}{c}\hfill -5\hfill \\ \hfill -5\hfill \\ \hfill -5\hfill \end{array}\right]$ is an eigenvector of $\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -2\hfill & \hfill 3\hfill \end{array}\right]$. What is the corresponding eigenvalue? (Enter your answer into the answer box.)

Correct!

Incorrect. Please try again.
Multiply $\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -2\hfill & \hfill 3\hfill \end{array}\right]$ by $\left[\begin{array}{c}\hfill -5\hfill \\ \hfill -5\hfill \\ \hfill -5\hfill \end{array}\right]$. The result should be a multiple of $\left[\begin{array}{c}\hfill -5\hfill \\ \hfill -5\hfill \\ \hfill -5\hfill \end{array}\right]$.
What are the eigenvalues of $\left[\begin{array}{ccc}\hfill 4\hfill & \hfill 7\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill -3\hfill & \hfill 8\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$? Exactly one option must be correct)
 a) $1$, $4$, $7$. b) $0$, $4$. c) $-3$, $2$, $4$. d) $-3$, $0$, $1$.

Choice (a) is incorrect
The eigenvalues of a triangular matrix are the entries on the main diagonal.
Choice (b) is incorrect
The eigenvalues of a triangular matrix are the entries on the main diagonal.
Choice (c) is correct!
Choice (d) is incorrect
The eigenvalues of a triangular matrix are the entries on the main diagonal.
Given that $1$ is an eigenvalue of $A=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 5\hfill & \hfill -6\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$, find the other two eigenvalues. Exactly one option must be correct)
 a) $2$ and $-3$. b) $-2$ and $3$. c) $0$ and $3$. d) $0$ and $3$.

Choice (a) is incorrect
$|A-\lambda I|=-{\lambda }^{3}+2{\lambda }^{2}+5\lambda -6$. Now take out a factor of $\left(\lambda -1\right)$.
Choice (b) is correct!
Choice (c) is incorrect
$|A-\lambda I|=-{\lambda }^{3}+2{\lambda }^{2}+5\lambda -6$. Now take out a factor of $\left(\lambda -1\right)$.
Choice (d) is incorrect
$|A-\lambda I|=-{\lambda }^{3}+2{\lambda }^{2}+5\lambda -6$. Now take out a factor of $\left(\lambda -1\right)$.
If $A=\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -1\hfill & \hfill 6\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 5\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill 0\hfill \end{array}\right]$, which one of the following is $|A-\lambda I|$? Exactly one option must be correct)
 a) $-{\lambda }^{3}+5{\lambda }^{2}+\lambda -14$ b) $-{\lambda }^{3}+5{\lambda }^{2}+\lambda -4$ c) $-{\lambda }^{3}+5{\lambda }^{2}-4\lambda$ d) $-{\lambda }^{3}+5{\lambda }^{2}-21\lambda +42$

Choice (a) is correct!
Choice (b) is incorrect
Try again.
Choice (c) is incorrect
Try again.
Choice (d) is incorrect
Try again.
Find the eigenvalues of $B=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill -4\hfill \\ \hfill -1\hfill & \hfill 4\hfill & \hfill 8\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill \end{array}\right]$. Exactly one option must be correct)
 a) $-1$, $1$, $4$. b) $-4$, $-1$, $1$. c) $-5$, $1$, $2$. d) $-2$, $1$, $5$.

Choice (a) is incorrect
Expanding $|B-\lambda I|$ by the first column gives

$\left(1-\lambda \right)\left[\left(4-\lambda \right)\left(-1-\lambda \right)-8\right]+2\left(1-\lambda \right).$ Now take out a factor of $\left(1-\lambda \right)$.
Choice (b) is incorrect
Expanding $|B-\lambda I|$ by the first column gives $\left(1-\lambda \right)\left[\left(4-\lambda \right)\left(-1-\lambda \right)-8\right]+2\left(1-\lambda \right).$ Now take out a factor of $\left(1-\lambda \right)$.
Choice (c) is incorrect
Expanding $|B-\lambda I|$ by the first column gives $\left(1-\lambda \right)\left[\left(4-\lambda \right)\left(-1-\lambda \right)-8\right]+2\left(1-\lambda \right).$ Now take out a factor of $\left(1-\lambda \right)$.
Choice (d) is correct!
Given that $5$ is an eigenvalue of $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill -4\hfill \\ \hfill -1\hfill & \hfill 4\hfill & \hfill 8\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill \end{array}\right]$, which of the following systems of equations should be solved to find the corresponding eigenvectors? Exactly one option must be correct)
 a) $\left[\begin{array}{ccc}\hfill -4\hfill & \hfill -3\hfill & \hfill -9\hfill \\ \hfill -6\hfill & \hfill -1\hfill & \hfill 3\hfill \\ \hfill -5\hfill & \hfill -4\hfill & \hfill -6\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right]$ b) $\left[\begin{array}{ccc}\hfill -4\hfill & \hfill 2\hfill & \hfill -4\hfill \\ \hfill -1\hfill & \hfill -1\hfill & \hfill 8\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -6\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right]=\left[\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right]$ c) $\left[\begin{array}{ccc}\hfill -4\hfill & \hfill 2\hfill & \hfill -4\hfill \\ \hfill -1\hfill & \hfill -1\hfill & \hfill 8\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -6\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right]$ d) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill -4\hfill \\ \hfill -1\hfill & \hfill 4\hfill & \hfill 8\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right]$ e) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill -4\hfill \\ \hfill -1\hfill & \hfill 4\hfill & \hfill 8\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 5\hfill \\ \hfill 5\hfill \\ \hfill 5\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Choice (e) is incorrect
A particular $3×3$ matrix $A$ has an eigenvalue of $-1$. The matrix $A+I$ reduces to $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$. Corresponding to the eigenvalue $-1$, all the eigenvectors of $A$ are non-zero vectors of the form: Exactly one option must be correct)
 a) $\left[\begin{array}{c}\hfill 2t\hfill \\ \hfill 0\hfill \\ \hfill t\hfill \end{array}\right],t\in ℝ$ b) $\left[\begin{array}{c}\hfill 2t\hfill \\ \hfill s\hfill \\ \hfill t\hfill \end{array}\right],s,\phantom{\rule{0.3em}{0ex}}t\in ℝ$ c) $\left[\begin{array}{c}\hfill t\hfill \\ \hfill 0\hfill \\ \hfill -2t\hfill \end{array}\right],t\in ℝ$ d) $\left[\begin{array}{c}\hfill t\hfill \\ \hfill s\hfill \\ \hfill 2t\hfill \end{array}\right],s,\phantom{\rule{0.3em}{0ex}}t\in ℝ$

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
An animal population with three age groups has Leslie matrix $\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 3.6\hfill \\ \hfill 0.8\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.6\hfill & \hfill 0\hfill \end{array}\right]$. Find the proportion of the population in each of the age groups once the population has stabilised. Exactly one option must be correct)
 a) Age group 1: $\frac{1}{3}$    Age group 2: $\frac{1}{3}$    Age group 3: $\frac{1}{3}$ b) Age group 1: $\frac{1}{6}$    Age group 2: $\frac{1}{3}$    Age group 3: $\frac{1}{2}$ c) Age group 1: $\frac{1}{4}$    Age group 2: $\frac{1}{2}$ Age group 3: $\frac{1}{4}$ d) Age group 1: $\frac{1}{2}$    Age group 2: $\frac{1}{3}$ Age group 3: $\frac{1}{6}$

Choice (a) is incorrect
Hint: Find the positive eigenvalue ($1.2$), and a corresponding eigenvector.
Choice (b) is incorrect
Hint: Find the positive eigenvalue ($1.2$), and a corresponding eigenvector.
Choice (c) is incorrect
Hint: Find the positive eigenvalue ($1.2$), and a corresponding eigenvector.
Choice (d) is correct!