Standard +0.3 This question tests standard eigenvalue concepts with a 3×3 matrix. The proof that e is an eigenvector of A² is straightforward algebraic manipulation. Finding eigenvalues requires computing a 3×3 determinant and solving a cubic, which is routine for Further Maths. The final part applies the proven result to a polynomial of B, requiring pattern recognition but no novel insight. Slightly above average difficulty due to the 3×3 computation and multi-part structure, but all techniques are standard FP1 material.
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
$$\mathbf { B } = \left( \begin{array} { l l l }
1 & 3 & 0 \\
2 & 0 & 2 \\
1 & 1 & 2
\end{array} \right)$$
Find the eigenvalues of \(\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix.
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
$$\mathbf { B } = \left( \begin{array} { l l l }
1 & 3 & 0 \\
2 & 0 & 2 \\
1 & 1 & 2
\end{array} \right)$$
Find the eigenvalues of $\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix.
\hfill \mbox{\textit{CAIE FP1 2013 Q7 [10]}}