Find the value of \(a\) for which the system of equations
$$\begin{gathered}
3 x + a y = 0
5 x - y = 0
x + 3 y + 2 z = 0
\end{gathered}$$
does not have a unique solution.
The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r }
3 & 0 & 0
5 & - 1 & 0
1 & 3 & 2
\end{array} \right)$$
Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
Use the characteristic equation of \(\mathbf { A }\) to show that
$$( \mathbf { A } + 6 \mathbf { I } ) ^ { 2 } = \mathbf { A } ^ { 4 } ( \mathbf { A } + b \mathbf { I } ) ^ { 2 }$$
where \(b\) is an integer to be determined.
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