## MATH1002 Quizzes

Quiz 11: Eigenvalues and eigenvectors
Question 1 Questions
Let $A=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]$. Which of the following statements are correct ? (Zero or more options can be correct)
 a) ${v}_{1}=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right)$ is an eigenvector of $A$ b) ${v}_{2}=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right)$is an eigenvector of $A$ c) ${v}_{3}=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -3\hfill \end{array}\right)$is an eigenvector of $A$ d) ${v}_{4}=\left(\begin{array}{c}\hfill -3\hfill \\ \hfill 1\hfill \end{array}\right)$is an eigenvector of $A$

There is at least one mistake.
For example, choice (a) should be True.
$A\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right)=3\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right)$, so $\mathbf{{v}_{1}}$ is a $3$–eigenvector of $A$
There is at least one mistake.
For example, choice (b) should be False.
$A\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 3\hfill \end{array}\right)$, so $\mathbf{{v}_{2}}$ is not an a eigenvector of $A$
There is at least one mistake.
For example, choice (c) should be True.
$A\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -3\hfill \end{array}\right)=-\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -3\hfill \end{array}\right)$, so $\mathbf{{v}_{3}}$ is a $-1$–eigenvector of $A$
There is at least one mistake.
For example, choice (d) should be False.
$A\left(\begin{array}{c}\hfill -3\hfill \\ \hfill 1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -5\hfill \\ \hfill -9\hfill \end{array}\right)$, so $\mathbf{{v}_{4}}$ is not an eigenvector of $A$
Correct!
1. True $A\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right)=3\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right)$, so $\mathbf{{v}_{1}}$ is a $3$–eigenvector of $A$
2. False $A\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 3\hfill \end{array}\right)$, so $\mathbf{{v}_{2}}$ is not an a eigenvector of $A$
3. True $A\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -3\hfill \end{array}\right)=-\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -3\hfill \end{array}\right)$, so $\mathbf{{v}_{3}}$ is a $-1$–eigenvector of $A$
4. False $A\left(\begin{array}{c}\hfill -3\hfill \\ \hfill 1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -5\hfill \\ \hfill -9\hfill \end{array}\right)$, so $\mathbf{{v}_{4}}$ is not an eigenvector of $A$
What are the eigenvalues of the matrix $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right]$ ? Exactly one option must be correct)
 a) 2 b) 0, -4 c) 0, 4 d) 2, -2

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$\begin{array}{rcll}\left|\begin{array}{cc}\hfill 2-\lambda \hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 2-\lambda \hfill \end{array}\right|& =& {\left(2-\lambda \right)}^{2}-4& \text{}\\ & =& {\lambda }^{2}-4\lambda +4-4& \text{}\\ & =& \lambda \left(\lambda -4\right)=0.& \text{}\end{array}$ The eigenvalues are therefore $\lambda =0$ and $\lambda =4$.
Choice (d) is incorrect
What are the eigenvalues of the matrix $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 7\hfill \\ \hfill -1\hfill & \hfill -3\hfill \end{array}\right]$ ? Exactly one option must be correct)
 a) $\frac{1+\sqrt{5}}{2}$, $\frac{1-\sqrt{5}}{2}$. b) $\frac{-1+\sqrt{3}}{2}$, $\frac{1-\sqrt{3}}{2}$. c) $\frac{-1+i\sqrt{3}}{2}$, $\frac{-1-i\sqrt{3}}{2}$. d) $\frac{5+i\sqrt{21}}{2}$, $\frac{5-i\sqrt{21}}{2}$.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$\begin{array}{rcll}\left|\begin{array}{cc}\hfill 2-\lambda \hfill & \hfill 7\hfill \\ \hfill -1\hfill & \hfill -3-\lambda \hfill \end{array}\right|& =& -\left(2-\lambda \right)\left(3+\lambda \right)+7& \text{}\\ & =& {\lambda }^{2}+\lambda +1=0.& \text{}\end{array}$ The eigenvalues are therefore $\lambda =\frac{-1±\sqrt{1-4}}{2}=\frac{-1±i\sqrt{3}}{2}$.
Choice (d) is incorrect
What is the characteristic equation of $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -6\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right]$ ? Exactly one option must be correct)
 a) ${\lambda }^{2}-3\lambda +8=0$ b) $detA={\lambda }^{2}+3\lambda +4$ c) ${\lambda }^{2}+3\lambda +4=0$ d) $detA={\lambda }^{2}-3\lambda +8$

Choice (a) is correct!
The characteristic equation of $A$ is $det\left(A-\lambda I\right)=0⇒\left|\begin{array}{cc}\hfill 1-\lambda \hfill & \hfill -6\hfill \\ \hfill 1\hfill & \hfill 2-\lambda \hfill \end{array}\right|$
$=\left(1-\lambda \right)\left(2-\lambda \right)+6={\lambda }^{2}-3\lambda +8=0$.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Let $A=\left[\begin{array}{cc}\hfill 3\hfill & \hfill 4\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right]$. Which of the following statements are correct? (Zero or more options can be correct)
 a) $\left(\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \end{array}\right)$ belongs to the $-1$–eigenspace of $A$ b) $\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \end{array}\right)$ belongs to the $5$–eigenspace of $A$ c) $\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right)$ belongs to the $5$–eigenspace of $A$ d) $\left(\begin{array}{c}\hfill -1\hfill \\ \hfill 1\hfill \end{array}\right)$ belongs to the $-1$–eigenspace of $A$

There is at least one mistake.
For example, choice (a) should be False.
As $A\left(\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 3\hfill \end{array}\right)\ne -1\left(\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \end{array}\right)$, $\left(\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \end{array}\right)$ does not belong to the $-1$–eigenspace of $A$
There is at least one mistake.
For example, choice (b) should be True.
As $A\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 10\hfill \\ \hfill 5\hfill \end{array}\right)=5\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \end{array}\right)$, $\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \end{array}\right)$ belongs to the $5$–eigenspace of $A$
There is at least one mistake.
For example, choice (c) should be False.
As $A\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -1\hfill \\ \hfill 1\hfill \end{array}\right)=-1\left(\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \end{array}\right)$, $\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right)$ belongs to the $-1$–eigenspace of $A$ and not the $5$–eigenspace of $A$
There is at least one mistake.
For example, choice (d) should be True.
As $A\left(\begin{array}{c}\hfill -1\hfill \\ \hfill 1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right)=-1\left(\begin{array}{c}\hfill -1\hfill \\ \hfill 1\hfill \end{array}\right)$, the vector $\left(\begin{array}{c}\hfill -1\hfill \\ \hfill 1\hfill \end{array}\right)$ belongs to in the $-1$–eigenspace of $A$.
Correct!
1. False As $A\left(\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 3\hfill \end{array}\right)\ne -1\left(\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \end{array}\right)$, $\left(\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \end{array}\right)$ does not belong to the $-1$–eigenspace of $A$
2. True As $A\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 10\hfill \\ \hfill 5\hfill \end{array}\right)=5\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \end{array}\right)$, $\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \end{array}\right)$ belongs to the $5$–eigenspace of $A$
3. False As $A\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -1\hfill \\ \hfill 1\hfill \end{array}\right)=-1\left(\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \end{array}\right)$, $\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right)$ belongs to the $-1$–eigenspace of $A$ and not the $5$–eigenspace of $A$
4. True As $A\left(\begin{array}{c}\hfill -1\hfill \\ \hfill 1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right)=-1\left(\begin{array}{c}\hfill -1\hfill \\ \hfill 1\hfill \end{array}\right)$, the vector $\left(\begin{array}{c}\hfill -1\hfill \\ \hfill 1\hfill \end{array}\right)$ belongs to in the $-1$–eigenspace of $A$.
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!
$A\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 2\hfill \\ \hfill 0\hfill \end{array}\right)=2\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}\right),\phantom{\rule{1em}{0ex}}A\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 2\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 2\hfill \\ \hfill 4\hfill \end{array}\right)=2\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 2\hfill \end{array}\right)$
$A\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill -1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -1\hfill \\ \hfill 1\hfill \\ \hfill -5\hfill \end{array}\right)$, $A\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 4\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 6\hfill \\ \hfill 6\hfill \\ \hfill 12\hfill \end{array}\right).$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
What are the eigenvalues of $\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -2\hfill & \hfill 3\hfill \end{array}\right]$ ? Exactly one option must be correct)
 a) 0,2,3 b) 1,2,3 c) 4,5 d) 2,3

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$\left|\begin{array}{ccc}\hfill 4-\lambda \hfill & \hfill -2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -\lambda \hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -2\hfill & \hfill 3-\lambda \hfill \end{array}\right|={\left(2-\lambda \right)}^{2}\left(3-\lambda \right)=0⇒\lambda =2$, $\lambda =3.$
Given that 1 is an eigenvalue of $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill -2\hfill \\ \hfill -1\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill \end{array}\right]$, what are the other eigenvalues ? Exactly one option must be correct)
 a) $1$, $2$ b) $-2$, $-1$ c) $-1$, $2$ d) $1$, $-2$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$|A-\lambda I|=\left|\begin{array}{ccc}\hfill 1-\lambda \hfill & \hfill -1\hfill & \hfill -2\hfill \\ \hfill -1\hfill & \hfill 2-\lambda \hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -1-\lambda \hfill \end{array}\right|=2-\lambda -2{\lambda }^{2}+{\lambda }^{3}=0$. Since $\left(1-\lambda \right)$ is a factor of $2-\lambda -2{\lambda }^{2}+{\lambda }^{3}$, $\left(1-\lambda \right)\left(a+b\lambda -{\lambda }^{2}\right)=2-\lambda -2{\lambda }^{2}+{\lambda }^{3}$ hence $a=2$, $b=1$ and the factors of $\left(2+\lambda -{\lambda }^{2}\right)$ are $\left(2-\lambda \right)$ and $\left(1+\lambda \right)$.
Choice (d) is incorrect
What is the characteristic equation of the matrix $\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -1\hfill & \hfill 6\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 6\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill 8\hfill \end{array}\right]$ ? Exactly one option must be correct)
 a) $36-45\lambda +13{\lambda }^{2}-{\lambda }^{3}=0$ b) $det\left(A-\lambda I\right)=36-45\lambda +13{\lambda }^{2}-{\lambda }^{3}$ c) $36-40\lambda +13{\lambda }^{2}-{\lambda }^{3}=0$ d) $det\left(A-\lambda I\right)=36-40\lambda +13{\lambda }^{2}-{\lambda }^{3}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$\begin{array}{rcll}\left|\begin{array}{ccc}\hfill 4-\lambda \hfill & \hfill -1\hfill & \hfill 6\hfill \\ \hfill 2\hfill & \hfill 1-\lambda \hfill & \hfill 6\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill 8-\lambda \hfill \end{array}\right|& =& \left(4-\lambda \right)\left[\left(1-\lambda \right)\left(8-\lambda \right)+6\right]+2\left(8-\lambda \right)-12+6\left[-2-2\left(1-\lambda \right)\right]& \text{}\\ & =& \left(4-\lambda \right)\left(2-\lambda \right)\left(7-\lambda \right)-10\left(2-\lambda \right)& \text{}\\ & =& \left(2-\lambda \right)\left[\left(4-\lambda \right)\left(7-\lambda \right)-10\right]& \text{}\\ & =& \left(2-\lambda \right)\left(2-\lambda \right)\left(9-\lambda \right)& \text{}\\ & =& 36-40\lambda +13{\lambda }^{2}-{\lambda }^{3}=0& \text{}\end{array}$
Choice (d) is incorrect
Let $\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 5\hfill & \hfill -6\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$. Which of the following statements are correct ?
(Zero or more options can be correct)
 a) $\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)$ is in the $2$-eigenspace of $A$ b) $\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 1\hfill \end{array}\right)$ is in the $3$-eigenspace of $A$ c) $\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)$ is in the $-2$-eigenspace of $A$ d) $\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 2\hfill \end{array}\right)$ is in the $-3$-eigenspace of $A$

There is at least one mistake.
For example, choice (a) should be False.
$A\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -8\hfill \\ \hfill 4\hfill \\ \hfill -2\hfill \end{array}\right)$, so $\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)$ belongs to the $-2$–eigenspace of $A$ (and not the $2$–eigenspace)
There is at least one mistake.
For example, choice (b) should be True.
$A\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 1\hfill \end{array}\right)=3\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 1\hfill \end{array}\right)$, so $\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 1\hfill \end{array}\right)$ belongs to the $3$–eigenspace of $A$
There is at least one mistake.
For example, choice (c) should be True.
$A\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)=-2\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)$, so $\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)$ belongs to the $-2$–eigenspace of $A$
There is at least one mistake.
For example, choice (d) should be False.
$A\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 2\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 21\hfill \\ \hfill 9\hfill \\ \hfill 3\hfill \end{array}\right)$, so $\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 2\hfill \end{array}\right)$ is not an eigenvector of $A$
Correct!
1. False $A\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -8\hfill \\ \hfill 4\hfill \\ \hfill -2\hfill \end{array}\right)$, so $\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)$ belongs to the $-2$–eigenspace of $A$ (and not the $2$–eigenspace)
2. True $A\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 1\hfill \end{array}\right)=3\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 1\hfill \end{array}\right)$, so $\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 1\hfill \end{array}\right)$ belongs to the $3$–eigenspace of $A$
3. True $A\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)=-2\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)$, so $\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)$ belongs to the $-2$–eigenspace of $A$
4. False $A\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 2\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 21\hfill \\ \hfill 9\hfill \\ \hfill 3\hfill \end{array}\right)$, so $\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 2\hfill \end{array}\right)$ is not an eigenvector of $A$