Further Mathematics 9231: Matrices Topical Past Paper Questions

Let $ \mathbf{A} = \begin{pmatrix} a & -1 \\ 5 & 1 \end{pmatrix} $, where $ a $ is a constant. It is given that 2 is an eigenvalue of A. (a) Find a and the other eigenvalue of A. (b) Find a matrix P and a diagonal matrix D such that $ A^{5} = \mathrm{PDP}^{-1} $. [5]

(a) Find the set of values of $a$ for which the system of equations $$ \begin{aligned}5x+ay&=3,\\20x-5y&=2,\\2x-3y+z&=1,\end{aligned} $$ has a unique solution and interpret this situation geometrically. [3] (b) Given that $ a = -\frac{5}{4} $, show that the system of equations in…

The matrix A is given by $$ \mathbf{A}=\begin{pmatrix}{{{-11}}}&{{{1}}}&{{{8}}} \\{{{0}}}&{{{-2}}}&{{{0}}} \\{{{-16}}}&{{{1}}}&{{{13}}}\end{pmatrix}. $$ (a) Show that $ \begin{pmatrix}1\\1\\1\end{pmatrix} $ is an eigenvector of A and state the corresponding eigenvalue. [2] (b) Sh…