The matrix \(\mathbf{M}\) is defined by
\(\mathbf{M}=\left(\begin{array}{ccc}
2 & m & 1 \\
0 & m & 7 \\
0 & 0 & 1
\end{array}\right),\)
where \(m \neq 0,1,2\).
(i) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{M}=\mathbf{P D P}^{-1}\).

(ii) Find \(\mathbf{M}^{7} \mathbf{P}\).

(a(i)) For \(B=\begin{pmatrix}k&0\\0&m\end{pmatrix}\), give full details of the transformation represented by \(B\) when \(m=1\).

(a(ii)) For \(B=\begin{pmatrix}k&0\\0&m\end{pmatrix}\), give full details of the transformation represented by \(B\) when \(m=k\).

(b) For \(A=\begin{pmatrix}0&1\\-1&1\\1&1\end{pmatrix}\), \(B=\begin{pmatrix}k&0\\0&m\end{pmatrix}\), and \(C=\begin{pmatrix}2&-1&1\\1&1&2\end{pmatrix}\), show that \(ABC\) is singular.

The matrix \(\mathbf{A}\) is defined by
\(\mathbf{A}=\left(\begin{array}{rrrr} 1 & -1 & -2 & -3 \\ -2 & 1 & 7 & 2 \\ -3 & 3 & 6 & \alpha \\ 7 & -6 & -17 & -17 \end{array}\right) .\)
(i) Show that if \(\alpha=9\) then the rank of \(\mathbf{A}\) is 2 , and find a basis for the null space of \(\mathbf{A}\) in this case.

(ii) Find the rank of \(\mathbf{A}\) when \(\alpha \neq 9\).