Find constant from eigenvalue condition

One of the eigenvalues of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 4 & 6 \\ 2 & - 4 & 2 \\ - 3 & 4 & a \end{array} \right)$$ is - 2 . Find the value of \(a\). Another eigenvalue of \(\mathbf { A }\) is - 5 . Find eigenvectors \(\mathbf { e } _ { 1 }\) and \(\mathbf { e } _ { 2 }\) corresponding to the eigenvalues - 2 and - 5 respectively. The linear space spanned by \(\mathbf { e } _ { 1 }\) and \(\mathbf { e } _ { 2 }\) is denoted by \(V\).

1. Prove that, for any vector \(\mathbf { x }\) belonging to \(V\), the vector \(\mathbf { A x }\) also belongs to \(V\).
2. Find a non-zero vector which is perpendicular to every vector in \(V\), and determine whether it is an eigenvector of \(\mathbf { A }\).

One of the eigenvalues of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r } 3 & - 4 & 2 \\ - 4 & \alpha & 6 \\ 2 & 6 & - 2 \end{array} \right)$$ is - 9 . Find the value of \(\alpha\). Find

1. the other two eigenvalues, \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\), of \(\mathbf { M }\), where \(\lambda _ { 1 } > \lambda _ { 2 }\),
2. corresponding eigenvectors for all three eigenvalues of \(\mathbf { M }\). It is given that \(\mathbf { x } = a \mathbf { e } _ { 1 } + b \mathbf { e } _ { 2 }\), where \(\mathbf { e } _ { 1 }\) and \(\mathbf { e } _ { 2 }\) are eigenvectors of \(\mathbf { M }\) corresponding to the eigenvalues \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\) respectively, and \(a\) and \(b\) are scalar constants. Show that \(\mathbf { M x } = p \mathbf { e } _ { 1 } + q \mathbf { e } _ { 2 }\), expressing \(p\) and \(q\) in terms of \(a\) and \(b\). \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. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. 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. }

1. A linear transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix

$$\mathbf { M } = \left( \begin{array} { c c } 5 & 1 \\ k & - 3 \end{array} \right)$$ where \(k\) is a constant.
Given that matrix \(\mathbf { M }\) has a repeated eigenvalue,

1. determine
   1. the value of \(k\)
   2. the eigenvalue.
2. Hence determine a Cartesian equation of the invariant line under \(T\).

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$\mathbf { A } = \left( \begin{array} { r r } 3 & k \\ - 5 & 2 \end{array} \right)$$ where \(k\) is a constant.
Given that there exists a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P }\) is a diagonal matrix where $$\mathbf { P } ^ { - 1 } \mathbf { A } \mathbf { P } = \left( \begin{array} { r r } 8 & 0 \\ 0 & - 3 \end{array} \right)$$

1. show that \(k = - 6\)
2. determine a suitable matrix \(\mathbf { P }\)