Standard +0.8 This is a multi-part Further Maths question requiring proof of eigenvalue properties, finding eigenvalues/eigenvectors of a 3×3 matrix (involving characteristic equation and solving), and applying the proven result. While systematic, the 3×3 characteristic polynomial and the synthesis in part (iii) elevate this above standard A-level, placing it moderately above average difficulty.
The vector \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\), with corresponding eigenvalue \(\lambda\), and is also an eigenvector of the matrix \(\mathbf { B }\), with corresponding eigenvalue \(\mu\). Show that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with corresponding eigenvalue \(\lambda \mu\).
Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { r r r }
0 & 1 & 3 \\
3 & 2 & - 3 \\
1 & 1 & 2
\end{array} \right) .$$
The matrix \(\mathbf { B }\), where
$$\mathbf { B } = \left( \begin{array} { r r r }
3 & 6 & 1 \\
1 & - 2 & - 1 \\
6 & 6 & - 2
\end{array} \right) ,$$
has eigenvectors \(\left( \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right) , \left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right)\). Find the eigenvalues of the matrix \(\mathbf { A B }\), and state corresponding eigenvectors.
(i) The vector $\mathbf { e }$ is an eigenvector of the matrix $\mathbf { A }$, with corresponding eigenvalue $\lambda$, and is also an eigenvector of the matrix $\mathbf { B }$, with corresponding eigenvalue $\mu$. Show that $\mathbf { e }$ is an eigenvector of the matrix $\mathbf { A B }$ with corresponding eigenvalue $\lambda \mu$.\\
(ii) Find the eigenvalues and corresponding eigenvectors of the matrix $\mathbf { A }$, where
$$\mathbf { A } = \left( \begin{array} { r r r }
0 & 1 & 3 \\
3 & 2 & - 3 \\
1 & 1 & 2
\end{array} \right) .$$
(iii) The matrix $\mathbf { B }$, where
$$\mathbf { B } = \left( \begin{array} { r r r }
3 & 6 & 1 \\
1 & - 2 & - 1 \\
6 & 6 & - 2
\end{array} \right) ,$$
has eigenvectors $\left( \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right) , \left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)$ and $\left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right)$. Find the eigenvalues of the matrix $\mathbf { A B }$, and state corresponding eigenvectors.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q11 EITHER}}