5:30
9231_s18_qp_11.pdf
8 UCIES 2018
\( 9231 / 11 / \mathrm{M} / 1 / 18 \)
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OR
It is given that \( \mathbf{e} \) is an eigenvector of the matrix \( \boldsymbol{A} \), with corresponding eigenvalue \( \lambda \).
(i) Write down another eigenvector of \( \mathbf{A} \) corresponding to \( \lambda \).
\( [1] \)
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(ii) Write down an eigenvector and corresponding eigenvalue of \( \mathbf{A}^{n} \), where \( n \) is a positive integer.
\( [2] \)
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Let \( \mathbf{A}=\left(\begin{array}{lll}3 & 0 & 0 \\ 2 & 7 & 0 \\ 4 & 8 & 1\end{array}\right) \)
(iii) Find a matrix \( \mathbf{P} \) and a diagonal matrix \( \mathbf{D} \) such that \( \mathbf{A}^{n}=\mathbf{P D P}^{-1} \),
\( [7] \)
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EUCIES 2018
\( 9231 / 11 / \mathrm{M} / / / 18 \)
9231_s18_ms_11.pdf
Answer
e.g. \( 2 \mathbf{e} \)
Eigenvector: \( \mathbf{e} \), Eigenvalue: \( \lambda^{n} \)