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I'm looking for the matrices $A$, $B$, $X$ such that $A^2 = B^3 = X$. Are there such matrices, and if so, how do I find them?

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    Um, A=C^3,B=C^2, X=C^6 for any matrix C? – 2017-02-20
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    For example, $X = I$ has every kind of root that you might want. – 2017-02-20
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    It turns out that every invertible matrix has some matrix with as a square/cube root. Every real invertible matrix has a real cube root. Some nilpotent matrices have no square/cube root at all. – 2017-02-20

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For any $n \in \mathbb{N}$, Let $Y$ be an $n\times n$ matrix.

Let $A = Y^3$ and $B = Y^2$ and $X = Y^6$. Then $$ A^2 = B^3 = X = Y^6$$