literally have no clue on how to do this one... :'(

Prove that if A and B are n x n matrices and if rank(A) = n and rank(B) = n then rank(AB) = n.

any help would be much appreciated here as i have a test tomorrow!

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- Mar 6th 2011, 11:22 AMsituationrank theory
literally have no clue on how to do this one... :'(

Prove that if A and B are n x n matrices and if rank(A) = n and rank(B) = n then rank(AB) = n.

any help would be much appreciated here as i have a test tomorrow! - Mar 6th 2011, 11:57 AMTheEmptySet
- Mar 6th 2011, 12:24 PMHallsofIvy
Another way- the "rank" of a linear transformation, A, from vector space V to itself is the dimension of A(V). Here, A is given by an n by n matrix so it is from $\displaystyle R^n$ to $\displaystyle R^n$. Since the rank of A is n, $\displaystyle A(R^n)= R^n$. Similarly, the rank of B is n so that $\displaystyle B(R^n)= R^n$. Putting those together, AB maps $\displaystyle R^n$ into all of $\displaystyle R^n$: $\displaystyle AB(R^n)= R^n$ and so AB has rank n.

- Mar 6th 2011, 01:31 PMsituation