Rank of the Product of Two Matrices rank{AB}=rank{B}-dim{N(A)∩R(B)}

A of size m-by-n, B of size n-by-q
rank{AB}=rank{B}-dim{N(A)∩R(B)}
Sylvester's inequality: Rank{AB} is greater than or equal to rank{A}+rank{B}-n
rank{A}+rank{B}-n less than or equal to rank{AB} less than or equal to min{rank{A},rank{B}}
Rank of a product is upperbounded by the minimum of the two ranks

The rank of the product formula can be used to prove Frobenius rank inequality related to the rank of the product of three matrices:
rank{XYZ} ≥ rank{XY} + rank{YZ} - rank{Y}