## H1002 Quizzes

Quiz 12: Eigenvalues and modelling
Question 1 Questions
The matrix $A=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right]$ has eigenvalues ${\lambda }_{1}=0$, ${\lambda }_{2}=4$ with corresponding eigenvectors ${v}_{1}=\left(\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \end{array}\right)$ and ${v}_{2}=\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \end{array}\right)$.
Which of the following matrices $T$ and $D$ satisfy the equation $AT=TD$ ?
 a) $T=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right]$, $D=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill -1\hfill \end{array}\right]$ b) $T=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right]$, $D=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 2\hfill \\ \hfill -1\hfill & \hfill 1\hfill \end{array}\right]$ c) $D=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right]$, $T=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill -1\hfill \end{array}\right]$ d) $D=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right]$, $T=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 2\hfill \\ \hfill -1\hfill & \hfill 1\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
For these two matrices $AT=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 8\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right]=TD$. In fact $D=\left[\begin{array}{cc}\hfill {\lambda }_{1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\lambda }_{2}\hfill \end{array}\right]$ and $T=\left(\begin{array}{cc}\hfill {v}_{1}\hfill & \hfill {v}_{2}\hfill \end{array}\right)$.
The matrix $A=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 5\hfill & \hfill -6\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$ has eigenvalues ${\lambda }_{1}=1$, ${\lambda }_{2}=3$, ${\lambda }_{3}=-2$ with corresponding eigenvectors ${v}_{1}=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 1\hfill \end{array}\right)$, ${v}_{2}=\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 3\hfill \\ \hfill 1\hfill \end{array}\right)$and ${v}_{3}=\left(\begin{array}{c}\hfill 4\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right)$.
Which of the following matrices $T$ and $D$ are such that $AT=TD$ ?
 a) $T=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -2\hfill \end{array}\right]$, $D=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 9\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$ b) $T=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -2\hfill \end{array}\right]$, $D=\left[\begin{array}{ccc}\hfill 4\hfill & \hfill 9\hfill & \hfill 1\hfill \\ \hfill -2\hfill & \hfill 3\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$ c) $D=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -2\hfill \end{array}\right]$, $T=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 9\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$ d) $D=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -2\hfill \end{array}\right]$, $T=\left[\begin{array}{ccc}\hfill 9\hfill & \hfill 1\hfill & \hfill 4\hfill \\ \hfill 3\hfill & \hfill 1\hfill & \hfill -2\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
$AT=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 27\hfill & \hfill -8\hfill \\ \hfill 1\hfill & \hfill 9\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \end{array}\right]=TD$.
Which are the matrices $T$ and $D$ such that $AT=TD$ when $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill 2\hfill \end{array}\right]$ ?
 a) $T=\left[\begin{array}{cc}\hfill 3\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right]$, $D=\left[\begin{array}{cc}\hfill 4\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ b) $T=\left[\begin{array}{cc}\hfill 3\hfill & \hfill 1\hfill \\ \hfill -2\hfill & \hfill 1\hfill \end{array}\right]$, $D=\left[\begin{array}{cc}\hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right]$ c) $T=\left[\begin{array}{cc}\hfill 3\hfill & \hfill 1\hfill \\ \hfill -2\hfill & \hfill 1\hfill \end{array}\right]$, $D=\left[\begin{array}{cc}\hfill 4\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right]$ d) $T=\left[\begin{array}{cc}\hfill 3\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right]$, $D=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is correct!
The eigenvalues of $A$ are ${\lambda }_{1}=-1$, ${\lambda }_{2}=4$ and the corresponding eigenvectors are ${v}_{1}=\left(\begin{array}{c}\hfill 3\hfill \\ \hfill -2\hfill \end{array}\right)$ and ${v}_{2}=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right)$. Hence $T=\left[\begin{array}{cc}\hfill 3\hfill & \hfill 1\hfill \\ \hfill -2\hfill & \hfill 1\hfill \end{array}\right]$ and $D=\left[\begin{array}{cc}\hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right]$.
Choice (c) is incorrect
Choice (d) is incorrect
The matrix $A=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 5\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill -13\hfill & \hfill -8\hfill & \hfill -6\hfill \end{array}\right]$ has eigenvectors ${v}_{1}=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \\ \hfill -1\hfill \end{array}\right)$, ${v}_{2}=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill -2\hfill \end{array}\right)$and ${v}_{3}=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill -3\hfill \end{array}\right)$.
Which are the matrices $T$ and $D$ such that $AT=TD$ ?
 a) $T=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill -2\hfill & \hfill -3\hfill \end{array}\right]$, $D=\left[\begin{array}{ccc}\hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ b) $T=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill -3\hfill & \hfill -1\hfill & \hfill -2\hfill \end{array}\right]$, $D=\left[\begin{array}{ccc}\hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ c) $T=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill -2\hfill & \hfill -3\hfill \end{array}\right]$, $D=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \end{array}\right]$ d) All of the above

Choice (a) is correct!
$A{v}_{1}=-{v}_{1}$, $A{v}_{2}=-2{v}_{2}$, $A{v}_{3}={v}_{3}$
$T=\left(\begin{array}{c}\hfill {v}_{1}\phantom{\rule{1em}{0ex}}{v}_{2}\phantom{\rule{1em}{0ex}}{v}_{3}\hfill \end{array}\right)$, $D=\left[\begin{array}{ccc}\hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Consider the matrix $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$. Find a matrix $T$ such that ${T}^{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}-1}AT=D$ for some diagonal matrix $D$.
 a) $T=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 9\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ b) $T=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 9\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ c) $T=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 6\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ d) No such $T$ exists.

Choice (a) is incorrect
It is true that $AT=TD$ where $D=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$, but det $T=0$, so ${T}^{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}-1}$ does not exist.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Note that $T=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 9\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ and $D=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$ are such that $AT=TD$, but det $T=0$, so ${T}^{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}-1}$ does not exist.
Let the Leslie matrix for a female population divided into 4 equal age groups be
$L=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 1.5\hfill & \hfill 2\hfill & \hfill 0.5\hfill \\ \hfill 0.4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.3\hfill & \hfill 0\hfill \end{array}\right]$
What is the fraction of the second age group which survives to the next generation ?
 a) 0.4 b) 0.5 c) 0.3 d) 1.5

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Let the Leslie matrix for a female population divided into 4 equal age groups be
$L=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 4.0\hfill & \hfill 3.9\hfill & \hfill 0.22\hfill \\ \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \end{array}\right].$
What is the per capita average number of female offspring for the 2nd age group?
 a) 0.2 b) 0.22 c) 3.9 d) 4

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
We are given that a female population is divided into 3 equal age groups and that the average no. of female offspring for the first, second and third age groups are 1$,$4.5 and 3.2 respectively. If the proportion of the first and second age groups which survive to the next age group is 0.9 and 0.7 respectively, what is the Leslie matrix which models this behaviour ?
 a) $L=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 4.5\hfill & \hfill 3.2\hfill \\ \hfill 0.9\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.7\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.7\hfill & \hfill 0\hfill \end{array}\right]$ b) $L=\left[\begin{array}{ccc}\hfill 0.9\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0.7\hfill & \hfill 4.5\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 3.2\hfill \end{array}\right]$ c) $L=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 4.5\hfill & \hfill 3.2\hfill \\ \hfill 0.9\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.7\hfill & \hfill 0\hfill \end{array}\right]$ d) There is not enough information provided.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
The female population of lemmings is divided into 4 equal age groups which are described by the following statistics: What is the Leslie matrix which models this behaviour ?
 a) $L=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2.4\hfill & \hfill 2\hfill & \hfill 0.7\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3.1\hfill & \hfill 3\hfill & \hfill 0.5\hfill \\ \hfill 0\hfill & \hfill 4.9\hfill & \hfill 4\hfill & \hfill 0\hfill \end{array}\right]$ b) $L=\left[\begin{array}{cccc}\hfill 2.4\hfill & \hfill 3.1\hfill & \hfill 4.9\hfill & \hfill 0.9\hfill \\ \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.7\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.5\hfill & \hfill 0\hfill \end{array}\right]$ c) $L=\left[\begin{array}{cccc}\hfill 2.4\hfill & \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 3.1\hfill & \hfill 0\hfill & \hfill 0.7\hfill & \hfill 0\hfill \\ \hfill 4.9\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.5\hfill \\ \hfill 0.9\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ d) Not enough information provided to answer the question.

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Two countries, A and B, have open door immigration policies between them. Let the population of countries $A$ and $B$ at the end of year $n$ be ${x}_{n}$ and ${y}_{n}$ respectively. The population of country A grows by 2% per year, excluding migration and the population of country $B$ grows by 5% per year excluding migration. It is observed that 10% of the increased population of country A migrates to country B and 7% of the increased population of country B migrates to country A each year. If $\left(\begin{array}{c}\hfill {x}_{n+1}\hfill \\ \hfill {y}_{n+1}\hfill \end{array}\right)=P\left(\begin{array}{c}\hfill {x}_{n}\hfill \\ \hfill {y}_{n}\hfill \end{array}\right)$, what is the matrix $P$ ?
 a) $P=\left[\begin{array}{cc}\hfill 0.9\hfill & \hfill 0.07\hfill \\ \hfill 0.1\hfill & \hfill 0.93\hfill \end{array}\right]$ b) $P=\left[\begin{array}{cc}\hfill 0.9\hfill & \hfill 0.07\hfill \\ \hfill 0.93\hfill & \hfill 0.1\hfill \end{array}\right]$ c) $P=\left[\begin{array}{cc}\hfill 0.918\hfill & \hfill 0.0735\hfill \\ \hfill 0.102\hfill & \hfill 0.9765\hfill \end{array}\right]$ d) $P=\left[\begin{array}{cc}\hfill 0.918\hfill & \hfill 0.0735\hfill \\ \hfill 0.9765\hfill & \hfill 0.102\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$\begin{array}{rcll}{x}_{n+1}& =& 0.9\left(1.02{x}_{n}\right)+0.07\left(1.05{y}_{n}\right)& \text{}\\ & =& 0.918{x}_{n}+0.0735{y}_{n}.& \text{}\\ {y}_{n+1}& =& 0.1\left(1.02{x}_{n}\right)+0.93\left(1.05{y}_{n}\right)& \text{}\\ & =& 0.102{x}_{n}+0.9765{y}_{n}.& \text{}\end{array}$
Choice (d) is incorrect