1.15 Eigenvalues and eigenvectorsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   $M=\begin{pmatrix}4&1\\ 2&3\end{pmatrix}$.

   (a)

   Find the characteristic polynomial of $M$.

   [1 mark]

   • A$\lambda^2+7\lambda+10$
   • B$\lambda^2-7\lambda-10$
   • C$\lambda^2-7\lambda+14$
   • D$\lambda^2-7\lambda+10$

   (b)

   Which vector is an eigenvector of $M$ with eigenvalue $5$?

   [1 mark]

   • A$\begin{pmatrix}1\\ -2\end{pmatrix}$
   • B$\begin{pmatrix}1\\ 1\end{pmatrix}$
   • C$\begin{pmatrix}2\\ 1\end{pmatrix}$
   • D$\begin{pmatrix}5\\ 1\end{pmatrix}$

   (c)

   Find an eigenvector of $M$ corresponding to the eigenvalue $2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $M=\begin{pmatrix}1&2\\ 3&0\end{pmatrix}$.

   (a)

   Find the eigenvalues of $M$.

   [1 mark]

   • A$3$ and $2$
   • B$1$ and $-6$
   • C$3$ and $-2$
   • D$-3$ and $2$

   (b)

   Which pair of matrices $P$ and $D$ satisfies $M=PDP^{-1}$, with $D$ diagonal?

   [1 mark]

   • A$P=\begin{pmatrix}1&2\\ 1&-3\end{pmatrix}$, $D=\begin{pmatrix}3&0\\ 0&-2\end{pmatrix}$
   • B$P=\begin{pmatrix}2&1\\ -3&1\end{pmatrix}$, $D=\begin{pmatrix}3&0\\ 0&-2\end{pmatrix}$
   • C$P=\begin{pmatrix}1&1\\ 2&-3\end{pmatrix}$, $D=\begin{pmatrix}3&0\\ 0&-2\end{pmatrix}$
   • D$P=\begin{pmatrix}1&2\\ 1&-3\end{pmatrix}$, $D=\begin{pmatrix}-2&0\\ 0&3\end{pmatrix}$

   (c)

   Write down the eigenvalues of $M^5$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The populations, in thousands, of two towns $A$ and $B$ in year $n$ are $a_n$ and $b_n$. Each year 20% of the people in town $A$ move to town $B$ and 10% of the people in town $B$ move to town $A$. Nobody else moves, so $\begin{pmatrix}a_{n+1}\\ b_{n+1}\end{pmatrix}=M\begin{pmatrix}a_n\\ b_n\end{pmatrix}$ with $M=\begin{pmatrix}0.8&0.1\\ 0.2&0.9\end{pmatrix}$.

   (a)

   Show that the eigenvalues of $M$ are $1$ and $0.7$.

   [3 marks]

   (b)

   Initially town $A$ has 60 thousand people and town $B$ has 30 thousand. Given that $M=PDP^{-1}$ with $P=\begin{pmatrix}1&1\\ 2&-1\end{pmatrix}$, $D=\begin{pmatrix}1&0\\ 0&0.7\end{pmatrix}$ and $P^{-1}=\frac13\begin{pmatrix}1&1\\ 2&-1\end{pmatrix}$, find the populations of the two towns after many years.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   On an island, the numbers of rabbits $R_n$ and foxes $F_n$ (both in hundreds) in year $n$ satisfy $\begin{pmatrix}R_{n+1}\\ F_{n+1}\end{pmatrix}=M\begin{pmatrix}R_n\\ F_n\end{pmatrix}$, where $M=\begin{pmatrix}1.4&-0.4\\ 0.4&0.4\end{pmatrix}$.

   (a)

   Find the eigenvalues of $M$ and a corresponding eigenvector for each.

   [6 marks]

   (b)

   Initially there are 400 rabbits and 500 foxes. Use your answer to part (a).
   (i) Write $\begin{pmatrix}4\\ 5\end{pmatrix}$ as a combination of the two eigenvectors.
   (ii) Hence find expressions for $R_n$ and $F_n$.
   (iii) Describe the long-term behaviour of the ratio $R_n:F_n$.

   [6 marks]

   Total for question 4: 12 marks

End of questions