How would I go about showing that for two matrices A, B, that the rank of (AB) (i.e. the rank of the matrix product of A and B) is less than or equal to the minimum of the rank of A and the rank of B?
That is:
rank (AB) ≤ min (rank(A), rank(B))
Thanks so much! :)
Michael T
Favorite Answer
I would start by exploring the issue for matrices in row-echelon form and see what you come up with. It makes sense it would be ≤ to the min since multiplying two matrices can't create new independent vectors and thus cannot increase the rank of the matrix you start with. What is more interesting is finding examples where this inequality is strict.