Find eigenvalues of 3×3 matrix

3 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 1 & 7 \\ 0 & 6 & 0 \\ 7 & 7 & 5 \end{array} \right) .$$

1. Find the eigenvalues of \(\mathbf { A }\).
2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-06_568_1614_294_262} The diagram shows the curve with equation \(\mathrm { y } = \ln \mathrm { x }\) for \(x \geqslant 1\), together with a set of ( \(N - 1\) ) rectangles of unit width.
3. By considering the sum of the areas of these rectangles, show that $$\ln N ! > N \ln N - N + 1 .$$
4. Use a similar method to find, in terms of \(N\), an upper bound for \(\operatorname { In } N\) !.

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 \(( 3 , - 6,0 )\).

1. Find Cartesian equations of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
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}
3. 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}
4. 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} If you use the following page to complete the answer to any question, the question number must be clearly shown.

4 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } - 11 & 1 & 8 \\ 0 & - 2 & 0 \\ - 16 & 1 & 13 \end{array} \right)$$

1. Show that \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\) and state the corresponding eigenvalue.
2. Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 19 \lambda - 30 = 0\) and hence find the other eigenvalues of \(\mathbf { A }\). \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-08_2717_35_106_2015} \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-09_2726_33_97_22}
3. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).

4 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } - 11 & 1 & 8 \\ 0 & - 2 & 0 \\ - 16 & 1 & 13 \end{array} \right)$$

1. Show that \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\) and state the corresponding eigenvalue.
2. Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 19 \lambda - 30 = 0\) and hence find the other eigenvalues of \(\mathbf { A }\). \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-08_2715_44_110_2006} \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-09_2726_33_97_22}
3. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).