Standard +0.3 This is a straightforward proof requiring direct application of eigenvalue definitions. The first part involves a simple algebraic manipulation (A²e = A(Ae) = A(λe) = λ²e), and the second part is routine substitution into a polynomial expression. Both parts require only basic understanding of eigenvalue properties with no novel insight or complex multi-step reasoning.
4 The matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue.
Given that one eigenvalue of \(\mathbf { A }\) is 3 , find an eigenvalue of the matrix \(\mathbf { A } ^ { 4 } + 3 \mathbf { A } ^ { 2 } + 2 \mathbf { I }\), justifying your answer.
4 The matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that $\mathbf { e }$ is an eigenvector of $\mathbf { A } ^ { 2 }$ and state the corresponding eigenvalue.
Given that one eigenvalue of $\mathbf { A }$ is 3 , find an eigenvalue of the matrix $\mathbf { A } ^ { 4 } + 3 \mathbf { A } ^ { 2 } + 2 \mathbf { I }$, justifying your answer.
\hfill \mbox{\textit{CAIE FP1 2008 Q4 [6]}}