One of the eigenvalues of the matrix $\mathbf { M }$, where

$$\mathbf { M } = \left( \begin{array} { r r r } 
3 & - 4 & 2 \\
- 4 & \alpha & 6 \\
2 & 6 & - 2
\end{array} \right)$$

is - 9 . Find the value of $\alpha$.

Find\\
(i) the other two eigenvalues, $\lambda _ { 1 }$ and $\lambda _ { 2 }$, of $\mathbf { M }$, where $\lambda _ { 1 } > \lambda _ { 2 }$,\\
(ii) corresponding eigenvectors for all three eigenvalues of $\mathbf { M }$.

It is given that $\mathbf { x } = a \mathbf { e } _ { 1 } + b \mathbf { e } _ { 2 }$, where $\mathbf { e } _ { 1 }$ and $\mathbf { e } _ { 2 }$ are eigenvectors of $\mathbf { M }$ corresponding to the eigenvalues $\lambda _ { 1 }$ and $\lambda _ { 2 }$ respectively, and $a$ and $b$ are scalar constants. Show that $\mathbf { M x } = p \mathbf { e } _ { 1 } + q \mathbf { e } _ { 2 }$, expressing $p$ and $q$ in terms of $a$ and $b$.

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\hfill \mbox{\textit{CAIE FP1 2015 Q11 OR}}