One of the eigenvalues of the matrix
$$\mathbf { A } = \left( \begin{array} { r r r }
1 & - 4 & 6
2 & - 4 & 2
- 3 & 4 & a
\end{array} \right)$$
is - 2 . Find the value of \(a\).
Another eigenvalue of \(\mathbf { A }\) is - 5 . Find eigenvectors \(\mathbf { e } _ { 1 }\) and \(\mathbf { e } _ { 2 }\) corresponding to the eigenvalues - 2 and - 5 respectively.
The linear space spanned by \(\mathbf { e } _ { 1 }\) and \(\mathbf { e } _ { 2 }\) is denoted by \(V\).
- Prove that, for any vector \(\mathbf { x }\) belonging to \(V\), the vector \(\mathbf { A x }\) also belongs to \(V\).
- Find a non-zero vector which is perpendicular to every vector in \(V\), and determine whether it is an eigenvector of \(\mathbf { A }\).