Standard +0.3 This is a straightforward two-part question on eigenvalues. The first part is a standard proof requiring only the definition of eigenvector (Ae = λe) and simple algebraic manipulation to show A²e = λ²e. The second part appears to be routine eigenvalue calculation. Both parts are textbook exercises requiring recall and standard techniques with no novel insight needed, making this slightly easier than average.
7 The square 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.
Find the eigenvalues of the matrix \(\mathbf { B }\), where
$$\mathbf { B } = \left( \begin{array} { l l l }
7 The square matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that $\mathbf { e }$ is an eigenvector of $\mathbf { A } ^ { 2 }$ and state the corresponding eigenvalue.
Find the eigenvalues of the matrix $\mathbf { B }$, where
$$\mathbf { B } = \left( \begin{array} { l l l }
\hfill \mbox{\textit{CAIE FP1 2013 Q7}}