Diagonalisation and the Cayley-Hamilton theoremEdexcel A-Level Further Maths: Mind map

Exam questions on Diagonalisation and the Cayley-Hamilton theorem

1. The matrix $\mathbf{A}=\begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$ has eigenvalues $2$ and $5$, with eigenvectors $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$ respectively. Let $\mathbf{P}=\begin{pmatrix} 1 & 1 \\ -2 & 1 \end{pmatrix}$.
   Write down a matrix $\mathbf{Q}$ such that $\mathbf{Q}^{-1}\mathbf{A}\mathbf{Q}=\begin{pmatrix} 5 & 0 \\ 0 & 2 \end{pmatrix}$.2 marks
2. The symmetric matrix $\mathbf{S}=\begin{pmatrix} 5 & 2 \\ 2 & 2 \end{pmatrix}$.
   Find an orthogonal matrix $\mathbf{Q}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{Q}^{\mathrm{T}}\mathbf{S}\mathbf{Q}=\mathbf{D}$.2 marks
3. The matrix $\mathbf{C}=\begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$.
   Find the characteristic equation of $\mathbf{C}$ and hence show that $\mathbf{C}^2=6\mathbf{C}-5\mathbf{I}$.3 marks