Possible values of rank of a matrix. Let $M$ be a $7×6$ real matrix. The entries of $M$ in the positions $(1, 3), (1, 4), (3, 3), (3, 4),$ and $(5, 4)$ are changed
to obtain another $7×6$ real matrix $N$. Suppose that the rank of $N$ is 4. What could be the rank of $M$?
I want to list all the possibilities. I can see that it should be less than $6$ and the rank is less than equal to the minimum of no. of.rows and cols. I did some examples, it seems like 2,3,4 are possible. But I am not getting a general idea. Kindly help me with this. Thank you. 
I am preparing for competitive exams and this is a question from one such paper. 
 A: The rank of a matrix is determined by the dimension of the span of the row vectors or the span of the columns vectors. Seeing the matrix as a list of row vectors then the condition of the problem say that you can change at most three row vectors (row one, three and five) , similarly seeing the matrix as a list of row columns the condition of the problem say that you can change at most two column vectors (column three and four).
Then you must see how much you can downgrade or upgrade the rank of the matrix studying how much you can make these row or column vectors dependent or independent. 
At most, without any analysis, you can diminish or increase the rank of the matrix by two, because it is enough to study just the column vectors (because the rank its the same as studying the row vectors), then it is enough to show that this could be possible giving some example of a matrix of rank four that, when changing the values of the coefficients of the exercise, it down to a matrix of rank two (this example is easy to find using the canonical column vectors that have zeros at all places except at a position) or it increases it rank to six.
