It is given that \(a\) is a positive constant.
(a) Show that the system of equations
\[\begin{aligned}
a x+(2 a+5) y+(a+1) z & =1 \\
-4 y & =2 \\
3 y-z & =3
\end{aligned}\]
has a unique solution and interpret this situation geometrically.

The matrix \(\mathbf{A}\) is given by
\[\mathbf{A}=\left(\begin{array}{ccc}
a & 2 a+5 & a+1 \\
0 & -4 & 0 \\
0 & 3 & -1
\end{array}\right)\]
(b) Show that the eigenvalues of \(\mathbf{A}\) are \(a,-1\) and -4 .

(c) Find a matrix \(\mathbf{P}\) such that
\[\mathbf{A}=\mathbf{P}\left(\begin{array}{rrr}
a & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -4
\end{array}\right) \mathbf{P}^{-1}\]
(d) Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{-1}\).