# Bound the Rank of BA Given Rank of AB Let `A` and `B` be real `4 x 4` matrices with `rank(AB) = 2`. Determine every possible value of `rank(BA)`. Prove the lower and upper bounds and show that each integer in the claimed range is attainable. ### Constraints & Assumptions - No invertibility, symmetry, or commutativity assumption is given. - A complete answer needs constructions or a general existence argument, not only rank inequalities. ### Clarifying Questions to Ask - Are explicit matrices required for the attainable ranks? - May standard rank-nullity and image/kernel formulas be used? ```hint Compare images with kernels Use `rank(AB) = rank(B) - dim(image(B) intersect kernel(A))`, and apply the analogous formula to `BA`. ``` ### What a Strong Answer Covers - Why rank four for `BA` is impossible. - Why rank zero is not excluded by `rank(AB) = 2`. - Constructions for all attainable intermediate ranks. - Correct use of rank, image, and kernel dimensions. ### Follow-up Questions - How does the range change for `n x n` matrices with `rank(AB) = r`? - What extra conclusion follows if either factor is invertible?