9 It is given that \(a\) is a positive constant.
- 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
$$\mathbf { A } = \left( \begin{array} { c c c }
a & 2 a + 5 & a + 1
0 & - 4 & 0
0 & 3 & - 1
\end{array} \right)$$ - Show that the eigenvalues of \(\mathbf { A }\) are \(a , - 1\) and - 4 .
- Find a matrix \(\mathbf { P }\) such that
$$\mathbf { A } = \mathbf { P } \left( \begin{array} { r r r }
a & 0 & 0
0 & - 1 & 0
0 & 0 & - 4
\end{array} \right) \mathbf { P } ^ { - 1 } .$$ - Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
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