Diagonalisation of symmetric matricesEdexcel International A Level Further Maths: Mind map

Exam questions on Diagonalisation of symmetric matrices

1. The symmetric matrix $\mathbf{S}=\begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}$.
   Show that eigenvectors of $\mathbf{S}$ corresponding to its two different eigenvalues are perpendicular.2 marks
2. The symmetric matrix $\mathbf{T}=\begin{pmatrix} 2 & 0 & 0 \\ 0 & 5 & 2 \\ 0 & 2 & 2 \end{pmatrix}$ can be reduced to diagonal form by an orthogonal matrix $\mathbf{P}$, so that $\mathbf{P}^T\mathbf{T}\mathbf{P}$ is diagonal.
   Find a normalised eigenvector of $\mathbf{T}$ corresponding to the eigenvalue $6$.2 marks
3. The symmetric matrix $\mathbf{S}=\begin{pmatrix} 1 & 2 \\ 2 & -2 \end{pmatrix}$.
   Find the eigenvalues of $\mathbf{S}$.3 marks