2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 3 & 4 \\ 2 & - 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 4 & 6 \\ 3 & - 5 \end{array} \right)\), and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Given that \(p \mathbf { A } + q \mathbf { B } = \mathbf { I }\), find the values of the constants \(p\) and \(q\).

## Question 2:

| Answer | Marks | Guidance |
|--------|-------|----------|
| State identity matrix is $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ | B1 | |
| $3p+4q=1,\quad -3p-5q=1,\quad 2p+3q=0$ | M1, A1 | Find a pair of simultaneous equations; Correct pair of distinct equations |
| $p=3$ and $q=-2$ | M1, A1 | Attempt to solve; Obtain correct answers |
| **[5]** | | |

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2 The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { r r } 3 & 4 \\ 2 & - 3 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { r r } 4 & 6 \\ 3 & - 5 \end{array} \right)$, and $\mathbf { I }$ is the $2 \times 2$ identity matrix. Given that $p \mathbf { A } + q \mathbf { B } = \mathbf { I }$, find the values of the constants $p$ and $q$.

\hfill \mbox{\textit{OCR FP1 2012 Q2 [5]}}