4 The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { r r r } 1 & 7 & 8
- 6 & 12 & 12
- 2 & 4 & 8 \end{array} \right)\).
- Show that the characteristic equation of \(\mathbf { P }\) is \(- \lambda ^ { 3 } + 21 \lambda ^ { 2 } - 126 \lambda + 216 = 0\).
You are given that the roots of this equation are 3,6 and 12 .
- Verify that \(\left( \begin{array} { r } 1
- 2
2 \end{array} \right)\) is an eigenvector of \(\mathbf { P }\), stating its associated eigenvalue. - The vector \(\left( \begin{array} { l } x
y
z \end{array} \right)\) is an eigenvector of \(\mathbf { P }\) with eigenvalue 6.
Given that \(z = 5\), find \(x\) and \(y\).
You are given that \(\mathbf { P }\) can be expressed in the form \(\mathbf { E D E } ^ { - 1 }\), where \(\mathbf { E } = \left( \begin{array} { r r r } 3 & 2 & 1
1 & 2 & - 2
1 & 1 & 2 \end{array} \right)\) and \(\mathbf { D }\) is a diagonal matrix. The characteristic equation of \(\mathbf { E }\) is \(- \lambda ^ { 3 } + 7 \lambda ^ { 2 } - 15 \lambda + 9 = 0\).
- Use the Cayley-Hamilton theorem to express \(\mathbf { E } ^ { - 1 }\) in terms of positive powers of \(\mathbf { E }\).
- Hence find \(\mathbf { E } ^ { - 1 }\).
- By identifying the matrix \(\mathbf { D }\) and using \(\mathbf { P } = \mathbf { E D E } ^ { - 1 }\), determine \(\mathbf { P } ^ { 4 }\).