Suppose I have a matrix A with integer elements and a matrix A mod p, where p is a prime number. In what circumstances rank(A)=rank(A mod p)?
Based on my understanding,
1) when det(A) is nonzero but not divisible by p, rank(A)=rank(A mod p) 2) when det(A) is zero, rank(A)=rank(A mod p)
If this is correct, how to prove it? Where can I find the relevant references?
Recently I am developing an algorithm where an integer matrix is computed, using modular arithmetic (mod p) can control the growth of integers in the matrix and thus improve the efficiency of my algorithm. The success of the algorithm requires rank of the matrix with mod does not change from the rank of the original matrix. This is where I will apply the problem. If someone can help me, I will really appreciate it.