Moderate -0.3 This is a straightforward FP1 matrices question testing basic operations: matrix multiplication (part i), inverse verification (part ii), and solving a matrix equation (part iii). All parts use standard algorithms with no conceptual challenges—slightly easier than average A-level due to the routine nature of the calculations, though the multi-part structure and FP1 context keep it near the mean.
You are given two matrices, **A** and **B**, where
$$\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \text{ and } \mathbf{B} = \begin{pmatrix} -1 & 2 \\ 2 & -1 \end{pmatrix}.$$
Show that \(\mathbf{AB} = m\mathbf{I}\), where \(m\) is a constant to be determined. [2]
You are given two matrices, **C** and **D**, where
$$\mathbf{C} = \begin{pmatrix} 2 & 1 & 5 \\ 1 & 1 & 3 \\ -1 & 2 & 2 \end{pmatrix} \text{ and } \mathbf{D} = \begin{pmatrix} -4 & 8 & -2 \\ -5 & 9 & -1 \\ 3 & -5 & 1 \end{pmatrix}.$$
Show that \(\mathbf{C}^{-1} = k\mathbf{D}\) where \(k\) is a constant to be determined. [2]
The matrices **E** and **F** are given by \(\mathbf{E} = \begin{pmatrix} k & k^2 \\ 3 & 0 \end{pmatrix}\) and \(\mathbf{F} = \begin{pmatrix} 2 \\ k \end{pmatrix}\) where \(k\) is a constant.
Determine any matrix **F** for which \(\mathbf{EF} = \begin{pmatrix} -2k \\ 6 \end{pmatrix}\). [5]
\begin{enumerate}[label=(\roman*)]
\item You are given two matrices, **A** and **B**, where
$$\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \text{ and } \mathbf{B} = \begin{pmatrix} -1 & 2 \\ 2 & -1 \end{pmatrix}.$$
Show that $\mathbf{AB} = m\mathbf{I}$, where $m$ is a constant to be determined. [2]
\item You are given two matrices, **C** and **D**, where
$$\mathbf{C} = \begin{pmatrix} 2 & 1 & 5 \\ 1 & 1 & 3 \\ -1 & 2 & 2 \end{pmatrix} \text{ and } \mathbf{D} = \begin{pmatrix} -4 & 8 & -2 \\ -5 & 9 & -1 \\ 3 & -5 & 1 \end{pmatrix}.$$
Show that $\mathbf{C}^{-1} = k\mathbf{D}$ where $k$ is a constant to be determined. [2]
\item The matrices **E** and **F** are given by $\mathbf{E} = \begin{pmatrix} k & k^2 \\ 3 & 0 \end{pmatrix}$ and $\mathbf{F} = \begin{pmatrix} 2 \\ k \end{pmatrix}$ where $k$ is a constant.
Determine any matrix **F** for which $\mathbf{EF} = \begin{pmatrix} -2k \\ 6 \end{pmatrix}$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 AS 2017 Q3 [9]}}