Challenging +1.2 This is a standard Further Maths eigenvalue/eigenvector question with a straightforward extension. Finding eigenvalues and eigenvectors of a 3×3 matrix is routine FP1 content. The twist involving B = A - kI requires recognizing that if λ is an eigenvalue of A, then (λ-k) is an eigenvalue of B, and (λ-k)³ is an eigenvalue of B³. This is a bookwork result that students are expected to know. The question is methodical but longer than average, placing it slightly above typical difficulty.
11 Find the eigenvalues of the matrix
$$\mathbf { A } = \left( \begin{array} { r r r }
- 1 & 1 & 4 \\
1 & 1 & - 1 \\
2 & 1 & 1
\end{array} \right)$$
and corresponding eigenvectors.
The matrix \(\mathbf { B }\) is defined by
$$\mathbf { B } = \mathbf { A } - k \mathbf { I } ,$$
where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix and \(k\) is a real number. Find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that
$$\mathbf { B } ^ { 3 } = \mathbf { P D } \mathbf { P } ^ { - 1 } .$$
11 Find the eigenvalues of the matrix
$$\mathbf { A } = \left( \begin{array} { r r r }
- 1 & 1 & 4 \\
1 & 1 & - 1 \\
2 & 1 & 1
\end{array} \right)$$
and corresponding eigenvectors.
The matrix $\mathbf { B }$ is defined by
$$\mathbf { B } = \mathbf { A } - k \mathbf { I } ,$$
where $\mathbf { I }$ is the $3 \times 3$ identity matrix and $k$ is a real number. Find a non-singular matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that
$$\mathbf { B } ^ { 3 } = \mathbf { P D } \mathbf { P } ^ { - 1 } .$$
\hfill \mbox{\textit{CAIE FP1 2007 Q11 [11]}}