Diagonalisation of matricesAQA A-Level Further Maths: Mind map

Exam questions on Diagonalisation of matrices

1. The matrix $M=\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$ has eigenvalues $2$ and $5$, with corresponding eigenvectors $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$.
   Taking $U=\begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}$, find $U^{-1}$.2 marks
2. The matrix $A=\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ can be written as $A=UDU^{-1}$, where $U=\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ and $D$ is a diagonal matrix.
   Explain why $A^4$ has the same eigenvectors as $A$, and state the eigenvalues of $A^4$.2 marks
3. The matrix $B=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 3 & 1 \\ 0 & 1 & 3 \end{pmatrix}$ has eigenvalues $1$, $2$ and $4$, with corresponding eigenvectors $\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, $\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$.
   Write down a matrix $U$ and a diagonal matrix $D$ such that $B=UDU^{-1}$.3 marks