Standard +0.3 This is a straightforward Further Maths question testing standard matrix operations and interpretation. Part (i) involves routine matrix inversion and solving BA=Y for A. Part (ii) requires recognizing that an enlargement followed by rotation gives matrix k·R(θ), then extracting k from the determinant and θ from the rotation matrix structure—all standard techniques with no novel problem-solving required.
$$\mathbf { B } = \left( \begin{array} { r r }
- 1 & 2 \\
3 & - 4
\end{array} \right) , \quad \mathbf { Y } = \left( \begin{array} { r r }
4 & - 2 \\
1 & 0
\end{array} \right)$$
Find \(\mathbf { B } ^ { - 1 }\).
The transformation represented by \(\mathbf { Y }\) is equivalent to the transformation represented by \(\mathbf { B }\) followed by the transformation represented by the matrix \(\mathbf { A }\).
Find \(\mathbf { A }\).
$$\mathbf { M } = \left( \begin{array} { r r }
- \sqrt { 3 } & - 1 \\
1 & - \sqrt { 3 }
\end{array} \right)$$
The matrix \(\mathbf { M }\) represents an enlargement scale factor \(k\), centre ( 0,0 ), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \(( 0,0 )\).
(a) Find the value of \(k\).
(b) Find the value of \(\theta\).
6.\\
(i)
$$\mathbf { B } = \left( \begin{array} { r r }
- 1 & 2 \\
3 & - 4
\end{array} \right) , \quad \mathbf { Y } = \left( \begin{array} { r r }
4 & - 2 \\
1 & 0
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Find $\mathbf { B } ^ { - 1 }$.
The transformation represented by $\mathbf { Y }$ is equivalent to the transformation represented by $\mathbf { B }$ followed by the transformation represented by the matrix $\mathbf { A }$.
\item Find $\mathbf { A }$.\\
(ii)
$$\mathbf { M } = \left( \begin{array} { r r }
- \sqrt { 3 } & - 1 \\
1 & - \sqrt { 3 }
\end{array} \right)$$
The matrix $\mathbf { M }$ represents an enlargement scale factor $k$, centre ( 0,0 ), where $k > 0$, followed by a rotation anti-clockwise through an angle $\theta$ about $( 0,0 )$.\\
(a) Find the value of $k$.\\
(b) Find the value of $\theta$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2014 Q6 [8]}}