/sci/ - Science & Math

Let A and B be two matrices such that AB is defined. If AB=A, does it automatically imply that B is the identity matrix of the order of the number of columns in A? There was a question saying that if AB=A and BA= B, prove that AA=A and BB=B. Pic related is the retarded solution they gave for it. I thought the most obvious way to proceed would be by saying that AB=A implies that B is the identity matrix. And hence BB must equal B, since I^2=I. Similarly, since BA=B, A=I. So, A=B=I. And AA= BB=I. Can there ever be a matrix B that is not the identity matrix such that AB=A? INB4 this website is 18+.