Question 31
Given a prime number $p$, let $A$ be a $p \times p$ matrix such that its entries are exactly $1, 2, \dots, p^2$ in some order. The following operation is allowed for a matrix: add one to each number in a row or a column, or subtract one from each number in a row or a column. The matrix $A$ is called “good” if one can take a finite series of such operations resulting in a matrix with all entries zero. Find the number of good matrices $A$.