# Infer BA When AB Is the Identity Matrices `A` and `B` satisfy `AB = I`. Explain what can be concluded about `BA`. The dimensions are not specified, so analyze both the case where `A` and `B` are square matrices of the same size and the compatible rectangular case. ### Constraints & Assumptions - Matrices are over the real numbers. - `I` is the identity matrix of the row dimension of `A`. ### Clarifying Questions to Ask - Are both matrices square, and if not, what are their dimensions? - Is the question asking whether `BA` is an identity, a projection, or merely constrained in rank? ```hint Interpret one-sided inverses For rectangular matrices, a right inverse need not also be a left inverse; examine what multiplying `BA` by itself does. ``` ### What a Strong Answer Covers - Invertibility and the determinant or rank argument in the square case. - The distinction between right and left inverses for rectangular matrices. - The idempotence and rank of `BA` when `AB = I` is rectangular. - A counterexample to the claim that `BA` must always be identity. ### Follow-up Questions - What are the eigenvalues of `BA` in the rectangular case? - Under what dimension condition can `AB = I` hold?