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I am trying to solve this question

$D$ is the matrix given by $D = \begin{pmatrix} h & 0 \\ 0 & k \end{pmatrix},$ where $h = 36$ and $k = -3$. What is the lower-right entry in the matrix $D^n$, where $n = 6$?

This is my attempt, but I am stuck half-way as it just doesn't seem right

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    There's also an error in the computation of the eigenvalues: $(-3 - \lambda) = 0$ if and only if $\lambda = -3$ and not $\lambda = 3$.2012-10-21

3 Answers 3

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What do you need the determinant for? It's way easier: check (by induction, for example) that

$\begin{pmatrix}h&0\\0&k\end{pmatrix}^n=\begin{pmatrix}h^n&0\\0&k^n\end{pmatrix}$

no matter what $\,h,k,n\,$ are

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    clearly this is way easier to do!2012-10-21
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For a diagonal matrix, the entries of $D^n$ are precisely the $n$th powers of the entries of $D$.

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And eigenvalues of a diagonal matrix are precisely diagonal elements by the definition.

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    Then this might better be a comment to the question.2012-10-21