Simultaneous Unimodularity of Positive Integer Matrices and Their Entrywise Cubes

We solve an open problem posed by Richard Guy in his book Unsolved Problems in Number Theory. We prove that, for every integer $n\geq10$, there exist infinitely many $n\times n$ integer matrices $A$ such that every entry is $\ge2$ and $\det A=\det A^{[3]}=1$, where $A^{[3]}$ denotes the entrywise cu