On an infinite number of solutions to the Diophantine equation $ x^{n}+y^{p}=z^{q}$ over the square integer matrices

Abstract: In this paper, we use some extension of the Cayley-Hamilton theorem to find a family of matrices with integer entries that satisfy the non-linear Diophantine equation $ x^{n}+y^{p}=z^{q}$ where $n,p$ and $q$ are arbitrary positive integers.