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Here is $3\times3$ matrix$$\begin{pmatrix} 1& -18& 0\\ 0 & 4& 0\\ -8& -13 & 9\end{pmatrix}$$ How can I find two different matrices so that $R^2=A$?

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    Reduce to RREF first? – 2017-01-19
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    Diagonalise? If $R$ works so does $-R$. – 2017-01-19

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Hint

Finding a square root is easy for a diagonal matrix. If you can write $A$ as $PDP^{-1}$ with $D$ a diagonal matrix and if $S^2 = D$, then $R=PSP^{-1}$ is a square root of $A$.

Your matrix is diagonalizable and has eigenvalues $1$, $4$ and $9$.

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    Interestingly, the eigenvalues are perfect squares, so this is a well crafted exercise, and your solution is for sure interesting and...doable. I just got myself a nice exercise for my poor hardworking students...+1 – 2017-01-19
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    Well they needn't be for this method to work but indeed, this probably isn't a coincidence... :-). – 2017-01-19
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    It's a nice application of diagonalization which I hadn't thought off.... – 2017-01-19
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    You're welcome @Ioanah. – 2017-01-20