uchun Eylerning matritsali teoremasi

The article investigates analogues of Euler's theorem for matrix traces and symmetric functions modulo 3 n p = , where p is a prime number. The main result proves the validity of the congruence ( ) ( ) tr( ) tr( ) mod n n n A A n  −  . Furthermore, the equivalence in the theory of symmetric functions and the divisibility properties of polynomial coefficients are analyzed using Newton's sums and multinomial coefficients. The proofs cover cases for both odd primes p and the specific case of 2 p =