(b) The triangle DEF in the x-y plane is transformed by \(\mathbf{A}\) onto triangle PQR. Given that the area of triangle DEF is 13 cm2, find the area of triangle PQR.
(c) Find the matrix \(\mathbf{B}\) such that \(\mathbf{AB} = \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix}\).
(d) Show that the origin is the only invariant point of the transformation in the x-y plane represented by \(\mathbf{A}\).
Solution
(a) The matrix \(\begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix}\) represents an enlargement with a scale factor of 6.
(b) The determinant of \(\mathbf{A}\) is \(\det(\mathbf{A}) = 3 \times 2 - 4 \times 2 = 6 - 8 = -2\). The area of triangle PQR is \(2 \times 13 = 26 \text{ cm}^2\).
(d) For \(\mathbf{A} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}\), we have \(\begin{pmatrix} 3x + 4y \\ 2x + 2y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}\). This leads to the equations \(3x + 4y = x\) and \(2x + 2y = y\). Solving these gives \(y = -2x\) and \(-6x = 0\), leading to \(x = 0, y = 0\). Thus, the origin is the only invariant point.
