Equivalence of Left and Right Inverse in Geometric Algebra

Question

I am asked by StackExchange to clarify if this is a duplicate question. This is not a duplicate of the question about matrix multiplication. This question is not (on its surface) a question about matrices.

I am beginning to learn Geometric Algebra.

Let $A, B$ be multivectors in Geometric Algebra $\mathbb{G}^n$. $AB = 1$ iff $BA = 1$. Why?

Progress: Let $e_{I}$ be a canonical basis vector, and $|I|$ is odd. Let $P$ be the projection operator onto this basis vector. Then $P[AB] = P[BA]$. The case of $|I|$ even gets messy. I do know for any two multivectors, the scalar part $S$ has the property $S[AB] = S[BA]$

Answer 1

A geometric algebra (these are basically the same as Clifford algebras) is a finite-dimensional algebra over a field. The implication $ab=1$ implies $ba=1$ always holds in such an algebra. One way to see this is that all such algebras have faithful representations as algebras of matrices, and $AB=I$ implies $BA=I$ is well-known for matrices.