9 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1
3 & a & 1
4 & 2 & 1 \end{array} \right)\).
- Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
- Hence find the values of \(a\) for which \(\mathbf { M } ^ { - 1 }\) does not exist.
- Determine whether the simultaneous equations
$$\begin{aligned}
& 6 x - 6 y + z = 3 k
& 3 x + 6 y + z = 0
& 4 x + 2 y + z = k
\end{aligned}$$
where \(k\) is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer. - Show that \(\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }\).
- Hence find an expression, in terms of \(n\), for
$$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
- Show that \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }\).