Standard +0.3 This is a straightforward Further Maths matrices question testing basic concepts: matrix multiplication/addition compatibility (routine recall), finding constants from a matrix equation (algebraic manipulation), deducing an inverse from AB=λI (direct application), and finding when a determinant equals zero (standard calculation with trigonometric simplification). All parts are textbook exercises requiring no novel insight, making it slightly easier than average even for Further Maths.
\begin{enumerate}[label=(\roman*)]
\item $\mathbf{A}$ is a 2 by 2 matrix and $\mathbf{B}$ is a 2 by 3 matrix.
Giving a reason for your answer, explain whether it is possible to evaluate
\begin{enumerate}[label=(\alph*)]
\item $\mathbf{AB}$
\item $\mathbf{A} + \mathbf{B}$
\end{enumerate}
[2]
\item Given that
$$\begin{pmatrix} -5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{pmatrix} \begin{pmatrix} 0 & 5 & 0 \\ 2 & 12 & -1 \\ -1 & -11 & 3 \end{pmatrix} = \lambda \mathbf{I}$$
where $a$, $b$ and $\lambda$ are constants,
\begin{enumerate}[label=(\alph*)]
\item determine
• the value of $\lambda$
• the value of $a$
• the value of $b$
\item Hence deduce the inverse of the matrix $\begin{pmatrix} -5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{pmatrix}$
\end{enumerate}
[3]
\item Given that
$$\mathbf{M} = \begin{pmatrix} 1 & 1 & 1 \\ 0 & \sin\theta & \cos\theta \\ 0 & \cos 2\theta & \sin 2\theta \end{pmatrix} \quad \text{where } 0 \leqslant \theta < \pi$$
determine the values of $\theta$ for which the matrix $\mathbf{M}$ is singular. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q4 [9]}}