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Given a matrix $A$ find a matrix $C$ such that $C^3$=$A$

I have stumbled upon the following question while studying for a test in linear algebra:

Find a matrix $X $ of $3 \times 3$ such that:

$X^3 = $ $ \left( \begin{array}{ccc} 0 & -1 & 1 \\ -1 & 0 & 1 \\ 0 & 0 & 1 \end{array} \right) $

I've never seen such questions before and I don't even know if I'm supposed to know how to solve it. Can anyone help me?

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marked as duplicate by user7530, rschwieb, hardmath, Austin Mohr, Old John Jan 21 '13 at 18:37

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Hint: diagonalize. –  Robert Israel Jan 21 '13 at 17:42
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1 Answer 1

For this particular question, you only need to verify that $\begin{pmatrix} 0 & -1 & 1 \\ -1 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix}^3=\begin{pmatrix} 0 & -1 & 1 \\ -1 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix}$. In general, if you want to find a matrix $X$ such that $X^3=A$ for some given $A$, you may try to diagonalize $A$ as $PDP^{-1}$ for some diagonal matrix $D$ and some invertible matrix $P$, and then set $X=PD^{1/3}P^{-1}$, where $D^{1/3}$ is the entrywise cubic root or $D$. You may use a computer algebra system to help you.

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You could explain the P=NP? problem using this answer. –  Christian Blatter Jan 21 '13 at 18:36
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