The vector \(\mathbf { e }\) is an eigenvector of each of the \(3 \times 3\) matrices \(\mathbf { A }\) and \(\mathbf { B }\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively. Justifying your answer, state an eigenvalue of \(\mathbf { A } + \mathbf { B }\). The matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 6 & - 1 & - 6
1 & 0 & - 2
3 & - 1 & - 3 \end{array} \right)$$ has eigenvectors \(\left( \begin{array} { l } 1
1
1 \end{array} \right) , \left( \begin{array} { r } 1
- 1
1 \end{array} \right) , \left( \begin{array} { l } 2
0
1 \end{array} \right)\). Find the corresponding eigenvalues. The matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { r r r } 8 & - 2 & - 8
2 & 0 & - 4
4 & - 2 & - 4 \end{array} \right) ,$$ also has eigenvectors \(\left( \begin{array} { l } 1
1
1 \end{array} \right) , \left( \begin{array} { r } 1
- 1
1 \end{array} \right) , \left( \begin{array} { l } 2
0
1 \end{array} \right)\), for which \(- 2,2,4\), respectively, are corresponding eigenvalues. The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \mathbf { A } + \mathbf { B } - 5 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. State the eigenvalues of \(\mathbf { M }\). Find matrices \(\mathbf { R }\) and \(\mathbf { S }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } ^ { 5 } = \mathbf { R D S }\).
[0pt] [You should show clearly all the elements of the matrices \(\mathbf { R } , \mathbf { S }\) and \(\mathbf { D }\).] \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }