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 $AT=TD$ ?

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 $AT=TD$ ?

Which are the matrices $T$ and $D$ such that $AT=TD$ when $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill 2\hfill \end{array}\right]$ ?

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 $AT=TD$ ?

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 $T^{-1}AT=D$ for some diagonal matrix $D$.

Let the Leslie matrix for a female population divided into 4 equal age groups be What is the fraction of the second age group which survives to the next generation ?

Let the Leslie matrix for a female population divided into 4 equal age groups be What is the average number of female offspring for the 2nd age group ?

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 ?

The female population of lemmings is divided into 4 equal age groups which are described by the following statisics: $$\begin{array}{ccc}\hfill \text{Age group}\hfill & \hfill \text{Average N}{}^{\text{o.}}\text{ offspring}\hfill & \hfill \text{\% surviving to next generation}\hfill \\ & & \\ \hfill 1\hfill & \hfill 2.4\hfill & \hfill 30\%\hfill \\ \hfill 2\hfill & \hfill 3.1\hfill & \hfill 70\%\hfill \\ \hfill 3\hfill & \hfill 4.9\hfill & \hfill 50\%\hfill \\ \hfill 4\hfill & \hfill 0.9\hfill & \hfill 0\%\hfill \end{array}$$ What is the Leslie matrix which models this behaviour ?

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 $P$ ?