Find the value of \(k\) for which the matrix
$$\mathbf { M } = \left( \begin{array} { r r r }
1 & - 1 & k
5 & 4 & 6
3 & 2 & 4
\end{array} \right)$$
does not have an inverse.
Assuming that \(k\) does not take this value, find the inverse of \(\mathbf { M }\) in terms of \(k\).
In the case \(k = 3\), evaluate
$$\mathbf { M } \left( \begin{array} { r }
- 3
3
1
\end{array} \right)$$
State the significance of what you have found in part (ii).
Find the value of \(t\) for which the system of equations
$$\begin{array} { r }
x - y + 3 z = t
5 x + 4 y + 6 z = 1
3 x + 2 y + 4 z = 0
\end{array}$$
has solutions. Find the general solution in this case and describe the solution geometrically.