Standard +0.3 This is a structured multi-part question on eigenvalues/eigenvectors that guides students through standard techniques. The proof is a straightforward algebraic manipulation, finding eigenvectors involves routine row reduction, and the final part applies the proven result to relate matrices B and C (where C = B - 3I). While it requires multiple steps and understanding of the theory, each component is a standard textbook exercise with clear signposting, making it slightly easier than average for Further Maths.
5 The matrix \(\mathbf { A }\) has an eigenvalue \(\lambda\) with corresponding eigenvector \(\mathbf { e }\). Prove that the matrix \(( \mathbf { A } + k \mathbf { I } )\), where \(k\) is a real constant and \(\mathbf { I }\) is the identity matrix, has an eigenvalue ( \(\lambda + k\) ) with corresponding eigenvector \(\mathbf { e }\).
The matrix \(\mathbf { B }\) is given by
$$\mathbf { B } = \left( \begin{array} { r r r }
2 & 2 & - 3 \\
2 & 2 & 3 \\
- 3 & 3 & 3
\end{array} \right) .$$
Two of the eigenvalues of \(\mathbf { B }\) are - 3 and 4 . Find corresponding eigenvectors.
Given that \(\left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\) is an eigenvector of \(\mathbf { B }\), find the corresponding eigenvalue.
Hence find the eigenvalues of \(\mathbf { C }\), where
$$\mathbf { C } = \left( \begin{array} { r r r }
- 1 & 2 & - 3 \\
2 & - 1 & 3 \\
- 3 & 3 & 0
\end{array} \right) ,$$
and state corresponding eigenvectors.
\((\mathbf{A}+k\mathbf{I})\mathbf{e} = \mathbf{A}\mathbf{e}+k\mathbf{I}\mathbf{e} = \lambda\mathbf{e}+k\mathbf{e} = (\lambda+k)\mathbf{e}\) \(\therefore (\mathbf{A}+k\mathbf{I})\) has eigenvalue \((\lambda+k)\) with eigenvector \(\mathbf{e}\)
M1A1
Part Mark: 2
Eigenvalues of \(\mathbf{B}\) are \(-3\) and \(4\) (given). Eigenvectors are \(\begin{pmatrix}1\\-1\\1\end{pmatrix}, \begin{pmatrix}1\\1\\0\end{pmatrix}\)
M1A1, A1
Part Mark: 3
Third eigenvalue is \(6\)
B1
Part Mark: 1
\(\mathbf{C} = \mathbf{B} - 3\mathbf{I}\) (stated or implied)
5 The matrix $\mathbf { A }$ has an eigenvalue $\lambda$ with corresponding eigenvector $\mathbf { e }$. Prove that the matrix $( \mathbf { A } + k \mathbf { I } )$, where $k$ is a real constant and $\mathbf { I }$ is the identity matrix, has an eigenvalue ( $\lambda + k$ ) with corresponding eigenvector $\mathbf { e }$.
The matrix $\mathbf { B }$ is given by
$$\mathbf { B } = \left( \begin{array} { r r r }
2 & 2 & - 3 \\
2 & 2 & 3 \\
- 3 & 3 & 3
\end{array} \right) .$$
Two of the eigenvalues of $\mathbf { B }$ are - 3 and 4 . Find corresponding eigenvectors.
Given that $\left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)$ is an eigenvector of $\mathbf { B }$, find the corresponding eigenvalue.
Hence find the eigenvalues of $\mathbf { C }$, where
$$\mathbf { C } = \left( \begin{array} { r r r }
- 1 & 2 & - 3 \\
2 & - 1 & 3 \\
- 3 & 3 & 0
\end{array} \right) ,$$
and state corresponding eigenvectors.
\hfill \mbox{\textit{CAIE FP1 2012 Q5 [9]}}