Ranks of Quotients, Remainders and $p$-Adic Digits of Matrices

For a prime p and a matrix A ∈ Zn×n, write A as A = p(A quo p)+ (A rem p) where the remainder and quotient operations are applied element-wise. Write the p-adic expansion of A as A = A[0] + pA[1] + p2A[2] + · · · where each A[i] ∈ Zn×n has entries between [0, p − 1]. Upper bounds are proven for the Z-ranks of A rem p, and A quo p. Also, upper bounds are proven for the Z/pZ-rank of A[i] for all i ≥ 0 when p = 2, and a conjecture is presented for odd primes.