Question

Wanted to ask how to prove "B'BA=B'BC is equivalent to BA=BC"....thx

Answer 1

No, because you can just cancel out the same matrix on the premise that it is invertible.

Answer 2

You can also multiply both sides by B, and then add some parentheses (associative property holds).
BB'BA=BB'BC
(BB')BA=(BB')BC

Answer 3

@crazydoglady yes, but you can only imply BA=BC if (BB') is invertible. And A and C have to be n x k, where n = number of columns of B and k is any number of cols.

Answer 4

How is the question written means "A1 iff A2" where A1:"B'BA=B'BC" and A2 : "BA=BC" as was mentioned by others here A2 implies A1 by pre-multiply both sides of BA=BC by B', however the other implication is not always true, should be true if as was mentioned by NGOG (BB') or B should be invertible