What AB = I Implies About BA

Suppose matrices A and B satisfy $AB = I$. State what can be inferred about BA. Since the dimensions are not given, treat both the square case, where A and B have the same size, and the rectangular case where the dimensions are compatible.

Constraints and Assumptions

• The matrices have real entries.
• I denotes the identity matrix whose size matches the number of rows of A.

Questions to Clarify

• Are A and B square, and if not, what are their dimensions?
• Is the goal to determine whether BA is an identity, a projection, or only restricted in rank?

Hint — Interpret one-sided inverses: With rectangular matrices, a right inverse does not have to be a left inverse; look at what happens when BA is multiplied by itself.

What a Strong Response Should Cover

• Invertibility, along with the determinant or rank argument, for the square case.
• How right and left inverses differ for rectangular matrices.
• The idempotence and rank of BA when the equation $AB = I$ involves rectangular matrices.
• A counterexample showing that BA does not always have to be the identity.

Follow-Up Questions

• What eigenvalues does BA have in the rectangular case?
• What dimension condition must hold for $AB = I$ to be possible?

Overview: Analyze what $AB = I$ implies about BA for square matrices of the same size and for rectangular matrices with compatible dimensions, including the projection case.