# Thread: Rank of 2 matrices Proof.

1. ## Rank of 2 matrices Proof.

Hello

Both of the below theorems are listed as properties 6 and 7 on the wikipedia page for the rank of a matrix.

I want to prove the following,

If $A$ is an M by n matrix and $B$ is a square matrix of rank n, then $rank(AB)$ = $rank (A)$.

Apparently this is a corollary to the theorem
If $A$ and $B$ are two matrices which can be multiplied, then $rank(AB)$ $\leq$ $min(rank (A), rank (B))$.

which I know how to prove. But I can't prove the first theorem. Any ideas?

2. ## Re: Rank of 2 matrices Proof.

What do you assume on $A$? If $A=0$ it won't work.

3. ## Re: Rank of 2 matrices Proof.

girdav, spotted and corrected typo.

4. ## Re: Rank of 2 matrices Proof.

You can do a reasoning using linear maps, and the fact that a bijective linear map preserves linear independence.

5. ## Re: Rank of 2 matrices Proof.

Originally Posted by girdav
You can do a reasoning using linear maps, and the fact that a bijective linear map preserves linear independence.
Do you know of any sources of a proof?