The matrix A is given by

\(A = \begin{pmatrix} k & 1 & 0 \\ 6 & 5 & 2 \\ -1 & 3 & -k \end{pmatrix}\),

where \(k\) is a real constant.

(a) Show that A is non-singular.

(b) Given that \(A^{-1} = \begin{pmatrix} 3 & 0 & -1 \\ 1 & 0 & 0 \\ -\frac{23}{2} & \frac{1}{2} & 3 \end{pmatrix}\), find the value of \(k\).