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?

One way to simplify the problem is to impose some further assumptions on $A$. For example, suppose we further assume that $A$ is singular. Then $A=uv^T$ for some entrywise vectors $u$ and $v$. The requirements that $A^3=A$ and $A\ne0$ entrywise are now equivalent to $(v^Tu)^2=1$ and $u,v\ne0$ entrywise, which are utterly easy to fulfil (when the characteristic of the underlying field is not $2$).
