Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 1 & 2
0 & 2 & 2
- 1 & 1 & 3 \end{array} \right)$$ The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is defined by \(\mathbf { x } \mapsto \mathbf { A x }\). Let \(\mathbf { e } , \mathbf { f }\) be two linearly independent eigenvectors of \(\mathbf { A }\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively, and let \(\Pi\) be the plane, through the origin, containing \(\mathbf { e }\) and \(\mathbf { f }\). By considering the parametric equation of \(\Pi\), show that all points of \(\Pi\) are mapped by T onto points of \(\Pi\). Find cartesian equations of three planes, each with the property that all points of the plane are mapped by T onto points of the same plane.