Links Between the Solvability of Matrix and Scalar Diophantine Equations

We investigate solutions to the matrix Diophantine equation Xⁿ + Yⁿ = n!Zⁿ within specific classes of 2 × 2 integer matrices. Firstly, we introduce a class of 2× 2 matrices G₂(d) and show that Xⁿ + Yⁿ = n!Zⁿ admits a solution in G₂(d) if and only if the classical Diophantine equation xⁿ + yⁿ = n!zⁿ has a solution in Q(√d) . We further present a constructive theorem for generating matrix solutions over a general number field K . Finally, we introduce a determinant-fixed class of matrices G₂(d, l) , and prove that the Fermat-type equation Xⁿ+Yⁿ=2Zⁿ has only the trivial solution in G₂(d, l) whenever $n>2$ .