Challenging +1.2 This is a structured multi-part question on eigenvalues/eigenvectors that guides students through standard proofs and computations. The first proof (showing λ^(-1) is an eigenvalue of A^(-1)) is a classic textbook result requiring straightforward algebraic manipulation. Finding eigenvectors and eigenvalues involves routine calculations. The final part requires understanding diagonalization and applying the earlier result, but follows a clear path. While it requires multiple techniques and careful bookkeeping across several parts, each individual step is standard Further Maths material with no novel insights required.
8 It is given that \(\lambda\) is an eigenvalue of the non-singular square matrix \(\mathbf { A }\), with corresponding eigenvector
e. Show that \(\lambda ^ { - 1 }\) is an eigenvalue of \(\mathbf { A } ^ { - 1 }\) for which \(\mathbf { e }\) is a corresponding eigenvector.
Deduce that \(\lambda + \lambda ^ { - 1 }\) is an eigenvalue of \(\mathbf { A } + \mathbf { A } ^ { - 1 }\).
It is given that 1 is an eigenvalue of the matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { r r r }
2 & 0 & 1 \\
- 1 & 2 & 3 \\
1 & 0 & 2
\end{array} \right)$$
Find a corresponding eigenvector.
It is also given that \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\) are eigenvectors of the matrix \(\mathbf { A }\). Find the corresponding eigenvalues.
Hence find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that
$$\left( \mathbf { A } + \mathbf { A } ^ { - 1 } \right) ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$
8 It is given that $\lambda$ is an eigenvalue of the non-singular square matrix $\mathbf { A }$, with corresponding eigenvector\\
e. Show that $\lambda ^ { - 1 }$ is an eigenvalue of $\mathbf { A } ^ { - 1 }$ for which $\mathbf { e }$ is a corresponding eigenvector.
Deduce that $\lambda + \lambda ^ { - 1 }$ is an eigenvalue of $\mathbf { A } + \mathbf { A } ^ { - 1 }$.
It is given that 1 is an eigenvalue of the matrix $\mathbf { A }$, where
$$\mathbf { A } = \left( \begin{array} { r r r }
2 & 0 & 1 \\
- 1 & 2 & 3 \\
1 & 0 & 2
\end{array} \right)$$
Find a corresponding eigenvector.
It is also given that $\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$ and $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)$ are eigenvectors of the matrix $\mathbf { A }$. Find the corresponding eigenvalues.\\
Hence find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that
$$\left( \mathbf { A } + \mathbf { A } ^ { - 1 } \right) ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$
\hfill \mbox{\textit{CAIE FP1 2014 Q8 [11]}}