# Rank of 2 matrices Proof.

• Jul 19th 2012, 11:24 AM
aukie
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 $\displaystyle A$ is an M by n matrix and $\displaystyle B$ is a square matrix of rank n, then $\displaystyle rank(AB)$ = $\displaystyle rank (A)$.

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

which I know how to prove. But I can't prove the first theorem. Any ideas?
• Jul 19th 2012, 12:35 PM
girdav
Re: Rank of 2 matrices Proof.
What do you assume on $\displaystyle A$? If $\displaystyle A=0$ it won't work.
• Jul 19th 2012, 12:58 PM
aukie
Re: Rank of 2 matrices Proof.
girdav, spotted and corrected typo.
• Jul 19th 2012, 01:00 PM
girdav
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.
• Jul 19th 2012, 01:14 PM
aukie
Re: Rank of 2 matrices Proof.
Quote:

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?