Possible values of rank of a matrix. [closed]

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.

• It also seems to me that all ranks between 2 and 6 are possible. A proof should state a) that the rank of M cannot be 0 or 1 and b) give examples of rank 2,3,4,5 and 6. It should be possible to give examples having entries 0 and 1 only. Mar 31, 2020 at 12:35

Since \begin{aligned} \operatorname{rank}(M)&=\operatorname{rank}(M-N+N)\le\operatorname{rank}(M-N)+\operatorname{rank}(N)\\ \operatorname{rank}(N)&=\operatorname{rank}(N-M+M)\le\operatorname{rank}(N-M)+\operatorname{rank}(M), \end{aligned} we have $$\operatorname{rank}(N)-\operatorname{rank}(N-M) \le\operatorname{rank}(M) \le\operatorname{rank}(M-N)+\operatorname{rank}(N).$$ By assumption, $$\operatorname{rank}(N)=4$$. Also, since all changes from $$M$$ to $$N$$ occur on two columns (namely, columns $$3$$ and $$5$$), we have $$\operatorname{rank}(N-M)=\operatorname{rank}(M-N)\le2$$. Therefore $$2\le\operatorname{rank}(M)\le6$$. To show that $$\operatorname{rank}(M)$$ can be any integer from $$2$$ to $$6$$, you need to exhibit some concrete examples. Below are some examples of changes of $$\{0,1\}$$-matrices that work over any field (not just the reals). Note that we are not changing only some of the entries $$m_{13},m_{14},m_{33},m_{34},m_{54}$$, but all of them.
Rank changes from $$2$$ to $$4$$: $$\pmatrix{ 0&0&\color{red}{0}&\color{red}{0}&0&0\\ 0&0&0&0&0&0\\ 0&0&\color{red}{0}&\color{red}{0}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&\color{red}{0}&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1} \to\pmatrix{ 0&0&\color{red}{1}&\color{red}{1}&0&0\\ 0&0&0&0&0&0\\ 0&0&\color{red}{1}&\color{red}{1}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&\color{red}{1}&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1}.$$ Rank changes from $$3$$ to $$4$$: $$\pmatrix{ 0&0&\color{red}{0}&\color{red}{1}&0&0\\ 0&0&0&0&0&0\\ 0&0&\color{red}{0}&\color{red}{1}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&\color{red}{0}&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1} \to\pmatrix{ 0&0&\color{red}{1}&\color{red}{0}&0&0\\ 0&0&0&0&0&0\\ 0&0&\color{red}{1}&\color{red}{0}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&\color{red}{1}&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1}.$$ Rank changes from $$4$$ to $$4$$: $$\pmatrix{ 0&0&\color{red}{1}&\color{red}{0}&0&0\\ 0&0&0&0&0&0\\ 0&0&\color{red}{0}&\color{red}{1}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&\color{red}{0}&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1} \to\pmatrix{ 0&0&\color{red}{0}&\color{red}{1}&0&0\\ 0&0&0&0&0&0\\ 0&0&\color{red}{1}&\color{red}{0}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&\color{red}{1}&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1}.$$ Rank changes from $$5$$ to $$4$$: $$\pmatrix{ 0&0&\color{red}{1}&\color{red}{0}&0&0\\ 0&1&0&0&0&0\\ 0&0&\color{red}{1}&\color{red}{0}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&\color{red}{1}&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1} \to\pmatrix{ 0&0&\color{red}{0}&\color{red}{1}&0&0\\ 0&1&0&0&0&0\\ 0&0&\color{red}{0}&\color{red}{1}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&\color{red}{0}&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1}.$$ Rank changes from $$6$$ to $$4$$: $$\pmatrix{ 0&0&\color{red}{1}&\color{red}{1}&0&0\\ 0&1&0&0&0&0\\ 0&0&\color{red}{1}&\color{red}{1}&0&0\\ 1&0&0&0&0&0\\ 0&0&0&\color{red}{1}&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1} \to\pmatrix{ 0&0&\color{red}{0}&\color{red}{0}&0&0\\ 0&1&0&0&0&0\\ 0&0&\color{red}{0}&\color{red}{0}&0&0\\ 1&0&0&0&0&0\\ 0&0&0&\color{red}{0}&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1}.$$
• Thank you for the wonderful answer. They have used all the entries $m_{13},m_{14},m_{33},m_{34},m_{54}$. So the person who set this question seems know about this and what could be his intuition about this problem? Kindly share your thoughts. Mar 31, 2020 at 16:06