Standard +0.3 This is a straightforward Core Pure 1 matrices question testing basic concepts: matrix multiplication/addition compatibility (routine recall), finding constants from matrix equation (simple arithmetic), deducing inverse from AB=λI (direct application of definition), and finding when determinant equals zero (standard calculation with trigonometric simplification). All parts are textbook exercises requiring no novel insight, making it slightly easier than average.
\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
\begin{itemize}
\item the value of $\lambda$
\item the value of $a$
\item the value of $b$
\end{itemize}
\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
$$\mathbf{M} = \begin{pmatrix} 1 & 1 & 1 \\ 0 & \sin\theta & \cos\theta \\ 0 & \cos 2\theta & \sin 2\theta \end{pmatrix} \quad \text{where } 0 \leq \theta < \pi$$
determine the values of $\theta$ for which the matrix $\mathbf{M}$ is singular.
[4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP1 2021 Q4 [9]}}