Find the value of \(a\) for which the system of equations
$$\begin{array} { r }
13 x + 18 y - 28 z = 0
- 4 x - a y + 8 z = 0
2 x + 6 y - 5 z = 0
\end{array}$$
does not have a unique solution.
The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r }
13 & 18 & - 28
- 4 & - 1 & 8
2 & 6 & - 5
\end{array} \right)$$
Find the eigenvalue of \(\mathbf { A }\) corresponding to the eigenvector \(\left( \begin{array} { l } 2 0 1 \end{array} \right)\).
Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\).
Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\) in terms of \(\mathbf { A }\).
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