<s>[INST] Find a matrix $A$ with no zero entries such that $A^3=A$ I took a standard $2 × 2$ matrix with entries $a, b, c, d$ and multiplied it out three times and tried to algebraically make it work, but that quickly turned into a algebraic mess. Is there an easier method to solve this?
 [/INST] If $ad-bc=1,$ with $a,b,c,d$ non-zero, you can use:
$$\begin{pmatrix}-bc&-bd\\ac&ad\end{pmatrix}$$
For example, $a=2,b=1,c=1,d=1$ gives:
$$\begin{pmatrix}-1&-1\\2&2\end{pmatrix}$$
I got this by taking:
$$\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}\begin{pmatrix}0&0\\0&1\end{pmatrix}\begin{pmatrix}a&b\\c&d\end{pmatrix}$$
Where $B=\begin{pmatrix}0&0\\0&1\end{pmatrix}$ satisfies $B^2=B$ and hence $B^3=B.$
You can also choose $B=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$ or $B=\begin{pmatrix}0&0\\0&-1\end{pmatrix}.$
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