Matrices: left inverse is also right inverse? [duplicate]

If A and B are square matrices, and AB=I, then I think it is also true that BA=I. In fact, this Wikipedia page says that this “follows from the associativity of matrix multiplication”. I assume there’s a nice simple one-line proof, but can’t seem to find it.

Nothing exotic, here — assume that the matrices have finite size and their elements are real numbers.

This isn’t homework (if that matters to you). My last homework assignment was about 50 years ago.

Answer

Since AB=I then B=B(AB)=(BA)B. Note from AB=I that 1=det so \det(B)\neq0.

So by (BA)B=B we have:

(BA-I)B=0. Since \det(B)\neq0 then B is not a 0 divisor. So BA=I

Attribution
Source : Link , Question Author : bubba , Answer Author : user71352

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