1.15 Eigenvalues and eigenvectorsIB Maths: Applications and Interpretation HL: Mind map

Exam questions on 1.15 Eigenvalues and eigenvectors

1. $M=\begin{pmatrix}4&1\\ 2&3\end{pmatrix}$.
   Find an eigenvector of $M$ corresponding to the eigenvalue $2$.2 marks
2. $M=\begin{pmatrix}1&2\\ 3&0\end{pmatrix}$.
   Write down the eigenvalues of $M^5$.2 marks
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 $M=\begin{pmatrix}0.8&0.1\\ 0.2&0.9\end{pmatrix}$.
   Show that the eigenvalues of $M$ are $1$ and $0.7$.3 marks