# Is there relationship between matrix nullity and rank of null matrix we can build

Give a matrix $$A$$ whose size e.g., $$m$$ x $$n$$, and $$m. I know the the null space of that matrix is $$X$$ such that $$A*X = 0$$. The nullity of matrix $$X$$ is e.g $$r$$. it means the null space I got is a matrix of size $$m$$ x $$r$$. As shown HERE, I can build the square $$m$$ x $$m$$ matrix $$Y$$ by taking the values of each column from the null space such that $$A*Y = 0$$.

My question, what is the maximum rank of $$Y$$ we can get by taking its value from the null space? Is it always equal to the nullity $$r$$? or it can be bigger?

The rank of a matrix $$M$$ is by definition the dimension of the vector space generated by its columns. If the columns belong to a space of dimension $$r$$, they generate a space of dimension $$\leq r$$, whence $$rank(M)\leq r$$.
• you mean maximum rank of matrix $Y$ in my case must be $r$ since I built $Y$ based on a matrix $X$ whose rank is $r$? May 20, 2021 at 10:43
• @Fatima_Ali Exactly. The maximum rank of $Y$ is the dimension of the null-space of $A$. May 20, 2021 at 10:44