Problem

The matrix $\mathbf{A}$ is defined by

$\mathbf{A} = \begin{pmatrix} 4 & -5 \\ -3 & 2 \end{pmatrix}$

The transformation represented by $\mathbf{A}$ maps triangle $T$ onto triangle $T'$

Given that the area of triangle $T$ is $23$ cm$^2$

(a) determine the area of triangle $T'$

The point $P$ has coordinates $(3p + 2, 2p - 1)$ where $p$ is a constant. The transformation represented by $\mathbf{A}$ maps $P$ onto the point $P'$ with coordinates $(17, -18)$

(b) Determine the value of $p$.

Given that

$\mathbf{B} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

(c) describe fully the single geometrical transformation represented by matrix $\mathbf{B}$

The transformation represented by matrix $\mathbf{A}$ followed by the transformation represented by matrix $\mathbf{C}$ is equivalent to the transformation represented by matrix $\mathbf{B}$

(d) Determine $\mathbf{C}$