Say $A$ is an $m\times n$ matrix having rank say $r$ and we obtain a matrix $B$ by changing one element of $A$, having rank say $s$. Then what can we say about the rank of $B$? Increase, decrease or remain same? How can we compare between $r$ and $s$?
Edit: Among the following which is/are possible? $$s\leq r+1,~ r-1\leq s$$~or~$$s=r-1?$$
For example, consider the matrix $$ A = \pmatrix{1&0\\0&0} $$ which has rank $1$. The new matrix $$ B = \pmatrix{1&0\\0&1} $$ has rank $2$. On the other hand, the new matrix $$ B = \pmatrix{1&1\\0&0} $$ also has rank $1$. On the other hand, the new matrix $$ B = \pmatrix{0&0\\0&0} $$ has rank $0$. So, there is no guarantee of what the rank will be, except to say that it will be within $1$ of the original.
As for your edit: the inequalities will always hold. $s = r-1$ is possible, but might not hold.