2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } 10 & 12 & - 8
- 1 & 2 & 4
3 & 6 & 2 \end{array} \right)\).
- In this question you must show detailed reasoning.
Show that the characteristic equation of \(\mathbf { A }\) is \(- \lambda ^ { 3 } + 14 \lambda ^ { 2 } - 56 \lambda + 64 = 0\).
- Use the Cayley-Hamilton theorem to determine \(\mathbf { A } ^ { - 1 }\).
A matrix \(\mathbf { E }\) and a diagonal matrix \(\mathbf { D }\) are such that \(\mathbf { A } = \mathbf { E D E } ^ { - 1 }\). The elements in the diagonal of \(\mathbf { D }\) increase from top left to bottom right.
- Determine the matrix \(\mathbf { D }\).