Standard +0.8 This is a Further Maths FP1 question on diagonalization requiring students to identify eigenvalues from an upper triangular matrix (straightforward), find eigenvectors by solving systems, construct P and D matrices, and apply the diagonalization to find A^n. While systematic, it requires multiple techniques and careful matrix algebra across several steps, placing it moderately above average difficulty.
10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { r r r }
- 2 & 1 & - 1 \\
0 & - 1 & 2 \\
0 & 0 & 1
\end{array} \right)$$
and find corresponding eigenvectors.
Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\), and hence find the matrix \(\mathbf { A } ^ { n }\), where \(n\) is a positive integer. [0pt]
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\(\mathbf{P} = \begin{pmatrix}1&1&0\\0&1&1\\0&0&1\end{pmatrix}, \mathbf{D} = \begin{pmatrix}-2&0&0\\0&-1&0\\0&0&1\end{pmatrix}\) or equivalent in correct order
10 Write down the eigenvalues of the matrix $\mathbf { A }$, where
$$\mathbf { A } = \left( \begin{array} { r r r }
- 2 & 1 & - 1 \\
0 & - 1 & 2 \\
0 & 0 & 1
\end{array} \right)$$
and find corresponding eigenvectors.
Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }$, and hence find the matrix $\mathbf { A } ^ { n }$, where $n$ is a positive integer.\\[0pt]
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\hfill \mbox{\textit{CAIE FP1 2016 Q10 [12]}}