What will be the rank of $M$ from the following information?

Question

Let $M$ be a $7\times6$ 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 $7\times6$ real matrix $N$ . Suppose that the rank of $N$ is $4$. What could be the rank of $M$ ?

I know that changing a single element of a matrix, rank can change at most one.

Answer 1

Hint: $N$ has the form $N = M + X$, where $X$ has the form $$ X = \pmatrix{ 0&0&x_{13}&x_{14}&0&0\\ 0&0&0&0&0&0\\ 0&0&x_{33}&x_{34}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&x_{54}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0}. $$ What could the rank of $X$ be?

Answer 2

By changing the entries you specified, you are changing the $3$rd and $4$th columns of the matrix. Therefore, the rank can change by at most $2$. If the ultimate rank is $4$, then the original rank is $4\pm 2$. Thus, the original rank is anything between $2$ and $6$.