Standard +0.8 This is a standard Cayley-Hamilton theorem application requiring finding the characteristic equation of a 3×3 matrix, then manipulating it to express A^4 in terms of lower powers. The triangular structure simplifies eigenvalue finding, but the algebraic manipulation to reach the specific form requires careful work across multiple steps, making it moderately challenging for Further Maths.
2 The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r }
- 1 & 2 & 12 \\
0 & 1 & 0 \\
0 & 0 & 3
\end{array} \right) .$$
Use the characteristic equation of \(\mathbf { A }\) to show that
$$\mathbf { A } ^ { 4 } = p \mathbf { A } ^ { 2 } + q \mathbf { l }$$
where \(p\) and \(q\) are integers to be determined.
2 The matrix $\mathbf { A }$ is given by
$$\mathbf { A } = \left( \begin{array} { r r r }
- 1 & 2 & 12 \\
0 & 1 & 0 \\
0 & 0 & 3
\end{array} \right) .$$
Use the characteristic equation of $\mathbf { A }$ to show that
$$\mathbf { A } ^ { 4 } = p \mathbf { A } ^ { 2 } + q \mathbf { l }$$
where $p$ and $q$ are integers to be determined.\\
\hfill \mbox{\textit{CAIE Further Paper 2 2021 Q2 [6]}}