FP1 June 2014 (R) Q3

(i) \[\mathbf{A} = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}\]

The matrix \(\mathbf{B}\) represents an enlargement, scale factor \(-2\), with centre the origin.

(ii) \[\mathbf{M} = \begin{pmatrix} 3 & k \\ -2 & 3 \end{pmatrix}, \quad \text{where } k \text{ is a positive constant.}\]

Triangle \(T\) has an area of 16 square units.

Triangle \(T\) is transformed onto the triangle \(T^{\prime}\) by the transformation represented by the matrix \(\mathbf{M}\).

Given that the area of the triangle \(T^{\prime}\) is 224 square units, find the value of \(k\). (3)