8 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) do not intersect and are both perpendicular to \(\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\). The line \(l\) intersects \(\Pi _ { 1 }\) at the point \(( 1,6,0 )\) and intersects \(\Pi _ { 2 }\) at the point \(( 3 , - 6,0 )\).
- Find Cartesian equations of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
- Express the vector equation of \(l\) in the form \(\left( \begin{array} { l } x
y
z \end{array} \right) = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be determined, and hence show that for points on \(l , \frac { 1 } { 2 } x + \frac { 1 } { 12 } y = 1\) and \(z = 0\).
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-16_2715_40_144_2008} - Show that the characteristic equation of \(\mathbf { A }\) is \(- \lambda ^ { 3 } + 3 \lambda ^ { 2 } + \frac { 7 } { 4 } \lambda = 0\) and hence find the eigenvalues of \(\mathbf { A }\).
The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { c c c }
1 & 2 & 3
1 & 2 & 3
\frac { 1 } { 2 } & \frac { 1 } { 12 } & 0
\end{array} \right)$$
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-17_194_1711_484_212} - Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }\), where \(n\) is a positive integer. [6]
\includegraphics[max width=\textwidth, alt={}]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_65_1581_335_322} ........................................................................................................................................
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_72_1579_511_324}
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_2718_35_144_2012}
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