Find the value of \(a\) for which the matrix
$$\mathbf { M } = \left( \begin{array} { r r r }
1 & 2 & 3
- 1 & a & 4
3 & - 2 & 2
\end{array} \right)$$
does not have an inverse.
Assuming that \(a\) does not have this value, find the inverse of \(\mathbf { M }\) in terms of \(a\).
Hence solve the following system of equations.
$$\begin{aligned}
x + 2 y + 3 z & = 1
- x + 4 z & = - 2
3 x - 2 y + 2 z & = 1
\end{aligned}$$
Find the value of \(b\) for which the following system of equations has a solution.
$$\begin{aligned}
x + 2 y + 3 z & = 1
- x + 6 y + 4 z & = - 2
3 x - 2 y + 2 z & = b
\end{aligned}$$
Find the general solution in this case and describe the solution geometrically.