Singular coefficient matrix and parallel lines

\(\mathbf{M}=\begin{pmatrix} k & 2 \\ 3 & k-1 \end{pmatrix}\), where \(k\) is a constant.

1. Find the values of \(k\) for which \(\mathbf{M}\) has no inverse.[3]
2. For the positive value of \(k\) found in (a), the equations \(\mathbf{M}\begin{pmatrix} x \\ y \end{pmatrix}=\begin{pmatrix} 5 \\ c \end{pmatrix}\) represent two straight lines. Explain, in terms of the lines, why there is no solution when \(c\ne5\) and infinitely many solutions when \(c=5\).[3]
3. Taking \(k=1\), use \(\mathbf{M}^{-1}\) to solve \(x+2y=4\) and \(3x=6\).[2]