(a) Find the set of values of \(a\) for which the system of equations
\(\begin{array}{c} 6 x+a y=3 \\ 2 x-y=1 \\ x+5 y+4 z=2 \end{array}\)
has a unique solution.

(b) Show that the system of equations in part (a) is consistent for all values of \(a\).

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 6 & 0 & 0 \\ 2 & -1 & 0 \\ 1 & 5 & 4 \end{array}\right)\)
(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((14 \mathbf{A}+24 \mathbf{I})^{2}=\mathbf{P D P}^{-1}\).

(d) Use the characteristic equation of \(\mathbf{A}\) to show that
\((14 \mathbf{A}+24 \mathbf{I})^{2}=\mathbf{A}^{4}(\mathbf{A}+b \mathbf{I})^{2}\)
where \(b\) is an integer to be determined.