Challenging +1.2 This question tests standard eigenvalue/eigenvector theory with straightforward applications. The proof is a direct algebraic manipulation using the definition of eigenvectors. The computational parts involve routine matrix-vector multiplication to find an eigenvalue, then applying the diagonalization formula with given eigenvalues/eigenvectors. While it requires understanding of multiple concepts (eigenvalues, diagonalization, matrix powers), each step follows standard procedures without requiring novel insight or complex problem-solving.
3 Show that if \(\lambda\) is an eigenvalue of the square matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the square matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector, then \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector.
The matrix
$$\mathbf { A } = \left( \begin{array} { r r r }
3 & - 1 & 0 \\
- 4 & - 6 & - 6 \\
5 & 11 & 10
\end{array} \right)$$
has \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\) as an eigenvector. Find the corresponding eigenvalue.
The other two eigenvalues of \(\mathbf { A }\) are 1 and 2, with corresponding eigenvectors \(\left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\) and \(\left( \begin{array} { r } 1 \\ 1 \\ - 2 \end{array} \right)\) respectively. The matrix \(\mathbf { B }\) has eigenvalues \(2,3,1\) with corresponding eigenvectors \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\), \(\left( \begin{array} { r } 1 \\ 1 \\ - 2 \end{array} \right)\) respectively. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 4 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\). [0pt]
[You are not required to evaluate \(\mathbf { P } ^ { - 1 }\).]
3 Show that if $\lambda$ is an eigenvalue of the square matrix $\mathbf { A }$ with $\mathbf { e }$ as a corresponding eigenvector, and $\mu$ is an eigenvalue of the square matrix $\mathbf { B }$ for which $\mathbf { e }$ is also a corresponding eigenvector, then $\lambda + \mu$ is an eigenvalue of the matrix $\mathbf { A } + \mathbf { B }$ with $\mathbf { e }$ as a corresponding eigenvector.
The matrix
$$\mathbf { A } = \left( \begin{array} { r r r }
3 & - 1 & 0 \\
- 4 & - 6 & - 6 \\
5 & 11 & 10
\end{array} \right)$$
has $\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)$ as an eigenvector. Find the corresponding eigenvalue.
The other two eigenvalues of $\mathbf { A }$ are 1 and 2, with corresponding eigenvectors $\left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)$ and $\left( \begin{array} { r } 1 \\ 1 \\ - 2 \end{array} \right)$ respectively. The matrix $\mathbf { B }$ has eigenvalues $2,3,1$ with corresponding eigenvectors $\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)$, $\left( \begin{array} { r } 1 \\ 1 \\ - 2 \end{array} \right)$ respectively. Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $( \mathbf { A } + \mathbf { B } ) ^ { 4 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$.\\[0pt]
[You are not required to evaluate $\mathbf { P } ^ { - 1 }$.]
\hfill \mbox{\textit{CAIE FP1 2008 Q3 [6]}}