Prove that if A is an invertible matrix and AB = BC then B = C.

[ha9981]

Prove that if A is an invertible matrix and AB = BC then B = C. I thought the way to approach it was to use A^-1 on the equality AB=BC but then I got stuck. I can't get it so that B is on one side and C alone is on the other.

My Try:
AB = BC
A^-1 AB = A^-1 BC
IB = A^-1BC


Also why does B = C not contradict the statement that "the cancellation law doesn't hold for matrices"?

 

[HallsofIvy]

Surely, that's not what you want to prove? It's not true. Take A= C and then, for any B, AB= BC because both are just AB.

You must have meant "If AB= AC then B= C". Just multiply both sides on the left by A[sup[-1[/sup].

And it does not contradict the statement "the cancellation law doesn't hold for matrices" because the cancellation law says that "If AB= AC for any non-zero A, then B= C". The theorem you are trying to prove requires that A be invertible. There are many non-zero matrices that are not invertible.