Standard +0.8 This is a substantial Further Maths question requiring matrix inversion with parameters, diagonalization from eigenvectors/eigenvalues, and application of Cayley-Hamilton theorem. While the techniques are standard FP2 content, the multi-part structure, algebraic manipulation with parameters, and the Cayley-Hamilton application to find matrix powers make this moderately challenging, though still following predictable patterns for this topic.
Given the matrix \(\mathbf { Q } = \left( \begin{array} { r r r } 2 & - 1 & k \\ 1 & 0 & 1 \\ 3 & 1 & 2 \end{array} \right)\) (where \(k \neq 3\) ), find \(\mathbf { Q } ^ { - 1 }\) in terms of \(k\).
Show that, when \(k = 4 , \mathbf { Q } ^ { - 1 } = \left( \begin{array} { r r r } - 1 & 6 & - 1 \\ 1 & - 8 & 2 \\ 1 & - 5 & 1 \end{array} \right)\).
The matrix \(\mathbf { M }\) has eigenvectors \(\left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \left( \begin{array} { r } - 1 \\ 0 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { l } 4 \\ 1 \\ 2 \end{array} \right)\), with corresponding eigenvalues \(1 , - 1\) and 3 respectively.
Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }\), and hence find the matrix \(\mathbf { M }\).
Write down the characteristic equation for \(\mathbf { M }\), and use the Cayley-Hamilton theorem to find integers \(a , b\) and \(c\) such that \(\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }\).
Section B (18 marks)
3 (i) Given the matrix $\mathbf { Q } = \left( \begin{array} { r r r } 2 & - 1 & k \\ 1 & 0 & 1 \\ 3 & 1 & 2 \end{array} \right)$ (where $k \neq 3$ ), find $\mathbf { Q } ^ { - 1 }$ in terms of $k$.\\
Show that, when $k = 4 , \mathbf { Q } ^ { - 1 } = \left( \begin{array} { r r r } - 1 & 6 & - 1 \\ 1 & - 8 & 2 \\ 1 & - 5 & 1 \end{array} \right)$.
The matrix $\mathbf { M }$ has eigenvectors $\left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \left( \begin{array} { r } - 1 \\ 0 \\ 1 \end{array} \right)$ and $\left( \begin{array} { l } 4 \\ 1 \\ 2 \end{array} \right)$, with corresponding eigenvalues $1 , - 1$ and 3 respectively.\\
(ii) Write down a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }$, and hence find the matrix $\mathbf { M }$.\\
(iii) Write down the characteristic equation for $\mathbf { M }$, and use the Cayley-Hamilton theorem to find integers $a , b$ and $c$ such that $\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }$.
Section B (18 marks)
\hfill \mbox{\textit{OCR MEI FP2 2008 Q3 [18]}}