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!
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.