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FM June 2023 p12 q04
4209
The matrix \(\mathbf{M}\) is given by \(\mathbf{M} = \begin{pmatrix} a & b^2 \\ c^2 & a \end{pmatrix}\), where \(a, b, c\) are real constants and \(b \neq 0\).
Show that \(\mathbf{M}\) does not represent a rotation about the origin.
Find the equations of the invariant lines, through the origin, of the transformation in the \(x-y\) plane represented by \(\mathbf{M}\).
It is given that \(\mathbf{M}\) represents the sequence of two transformations in the \(x-y\) plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, x-axis fixed, with \((0, 1)\) mapped to \((5, 1)\). Find \(\mathbf{M}\).
The triangle \(DEF\) in the \(x-y\) plane is transformed by \(\mathbf{M}\) onto triangle \(PQR\). Given that the area of triangle \(DEF\) is \(12 \text{ cm}^2\), find the area of triangle \(PQR\).
Solution
(a) For \(\mathbf{M}\) to represent a rotation, it must be of the form \(\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\). Comparing with \(\mathbf{M} = \begin{pmatrix} a & b^2 \\ c^2 & a \end{pmatrix}\), we have \(b^2 = -c^2\), which is impossible since \(b\) and \(c\) are real and \(b \neq 0\).
(b) The transformation of \(\begin{pmatrix} x \\ y \end{pmatrix}\) by \(\mathbf{M}\) gives \(\begin{pmatrix} ax + b^2y \\ c^2x + ay \end{pmatrix}\). For invariant lines through the origin, \(y = mx\) gives \(c^2x + amx = m(ax + b^2y)\). Simplifying, \(c^2 + am = ma + b^2m^2\) leads to \(c^2 = b^2m^2\), giving \(y = \frac{c}{b} x\) and \(y = -\frac{c}{b} x\).
(d) The area of \(PQR\) is \(12 \times \det(\mathbf{M})\). \(\det(\mathbf{M}) = 5 \times 5 - 0 \times 25 = 25\). Therefore, the area is \(12 \times 25 = 300 \text{ cm}^2\).
