Find the set of values of \(a\) for which the system of equations
$$\begin{array} { c l }
6 x + a y & = 3
2 x - y & = 1
x + 5 y + 4 z & = 2
\end{array}$$
has a unique solution.
Show that the system of equations in part (a) is consistent for all values of \(a\).
The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r }
6 & 0 & 0
2 & - 1 & 0
1 & 5 & 4
\end{array} \right)$$
Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( 14 \mathbf { A } + 24 \mathbf { I } ) ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
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.
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