| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | November |
| Marks | 4 |
| Topic | Matrices |
| Type | Properties of matrix operations |
| Difficulty | Moderate -0.8 This question tests basic matrix properties: (i) requires knowing that non-singular means determinant ≠ 0, a direct calculation; (ii) uses the standard result (AB)^(-1) = B^(-1)A^(-1) and matrix multiplication. Both parts are routine applications of fundamental matrix theory with no problem-solving insight required, making this easier than average even for Further Maths. |
| Spec | 4.03o Inverse 3x3 matrix4.03p Inverse properties: (AB)^(-1) = B^(-1)*A^(-1) |
For real values of $t$, the non-singular matrices $\mathbf{A}$ and $\mathbf{B}$ are such that
$$\mathbf{A}^{-1} = \begin{pmatrix} t & 5 \\ 2 & 8 \end{pmatrix} \quad \text{and} \quad \mathbf{B}^{-1} = \begin{pmatrix} 2 & -t \\ 3 & -1 \end{pmatrix}.$$
\begin{enumerate}[label=(\roman*)]
\item Determine the values which $t$ cannot take. [2]
\item Without finding either $\mathbf{A}$ or $\mathbf{B}$, determine $(\mathbf{AB})^{-1}$ in terms of $t$. [2]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q1 [4]}}