The matrices A, B and C are given by

\(A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{pmatrix}, \ B = \begin{pmatrix} 0 & -2 \\ -1 & 3 \\ 0 & 0 \end{pmatrix} \text{ and } C = \begin{pmatrix} -2 & -1 \\ 1 & 1 \end{pmatrix}.\)

(a) Show that \(CAB = \begin{pmatrix} 3 & -7 \\ -9 & 3 \end{pmatrix}.\) [3]

(b) Find the equations of the invariant lines, through the origin, of the transformation in the \(x-y\) plane represented by \(CAB.\) [5]

Let \(M = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}.\)

(c) Give full details of the transformation represented by \(M.\) [2]

(d) Find the matrix \(N\) such that \(NM = CAB.\) [3]