Wait, we don't know if Fermat's theorem or Euler's theorem applies for matrices right? Isn't it more about Binary Exponentiation? What is the modulo of a matrix?

Adding more: I think you can't say that $$$ A^{a^{b}} \mod p $$$ gives same answer as $$$A^{ ( a^b \mod {p-1} ) } \mod p $$$, where $$$ A \mod p $$$ is taking element wise modulo of all entries in $$$A$$$.

Since multiplying matrices only has multiplication and addition operations, we can definitely agree that $$$ (A*B) \mod p $$$ is same as $$$ ( A \mod p ) * ( B \mod p ) \mod p $$$. But exponentiation is different, to take modulo of the exponent by $$$phi(p)$$$, you need to show that $$$ A^{phi(p)} $$$ will be identity matrix.