CAIE FP1 2019 November — Question 8

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2019
SessionNovember
TopicInvariant lines and eigenvalues and vectors

8 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left( \begin{array} { c c c } 2 & m & 1
0 & m & 7
0 & 0 & 1 \end{array} \right) ,$$ where \(m \neq 0,1,2\).
  1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } = \mathbf { P D P } ^ { - 1 }\).
  2. Find \(\mathbf { M } ^ { 7 } \mathbf { P }\).