# Rank, nullity and the number of rows of a matrix

I have this question here:

Let $$A$$ be a matrix with $$10$$ columns, $$dim$$ $$Null(A)=5$$ and $$dim$$ $$Null(A^T)=3$$. How many rows does $$A$$ have?

$$a)$$ $$8$$

$$b)$$ $$3$$

$$c)$$ $$5$$

$$d)$$ $$10$$

$$e)$$ This cannot be determined from the information given

I tried doing it. I know that:

$$rank(A)+Nullity(A)=10$$

So that means that

$$rank(A)+5=10$$

$$rank(A)=5$$

However, I am not really sure how that helps me find the number of rows. I know that $$dim$$ $$Null(A^T)=3$$ but how do I incorporate that into this?

Thanks!

## 1 Answer

$$\textbf{Note:}$$ $$\text{Rank(A) = dim(rowsp(A)) = dim(colsp(A))}$$

$$\text{Nullity(A^T) + dim(rowsp(A)) = Number of rows of A}$$

$$\text{Nullity(A) + dim(colsp(A)) = Number of columns of A}$$

• Yup I see how that works. I know that the 2nd equation is true but why is $\text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of$A$}$ true? How do you obtain that? – Future Math person Dec 19 '18 at 6:31
• $\text{Nullity($A^T$)}$ is the number of zero rows in RREF of $A^T$ and $\text{dim(rowsp($A$))}$ is the number of pivot rows. – Yadati Kiran Dec 19 '18 at 6:36
• In other words consider the third equation for $A^T$ then rows become columns and columns become rows. Then we have $\text{Nullity($A^T$) + dim(colsp($A^T$)) = Number of columns of$A^T$=Number of rows of$A$}$ – Yadati Kiran Dec 19 '18 at 6:42
• I get it now! Thanks so much! – Future Math person Dec 19 '18 at 6:49