5.
$$\mathbf { A } = \left( \begin{array} { r r r }
a & a & 1
- a & 4 & 0
4 & a & 5
\end{array} \right) \quad \text { where } a \text { is a positive constant }$$
- Determine the exact value of \(a\) for which the matrix \(\mathbf { A }\) is singular.
Given that 2 is an eigenvalue of \(\mathbf { A }\)
- determine
- the value of \(a\)
- the other two eigenvalues of \(\mathbf { A }\)
A normalised eigenvector for the eigenvalue 2 is \(\left( \begin{array} { c } \frac { 1 } { \sqrt { 6 } }
\frac { 1 } { \sqrt { 6 } }
- \frac { 2 } { \sqrt { 6 } } \end{array} \right)\)
- Determine a normalised eigenvector for each of the other eigenvalues of \(\mathbf { A }\)
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