An unexpected result: AC=BC dose not imply that A=B . Explain why?

Question

An unexpected result: AC=BC dose not imply that A=B . Explain why

Answer 1

The only case that I could think of that A is not equal to B where AC=BC is when C=0. Other than that A must equal B for the expression to be valid

Answer 2

I think this relates to inductive vs. deductive reasoning.

If you start with the premise that A = B, then you are guaranteed that AC = BC (provided A, B, and C are real).

However, if you start with the premise that AC = BC, then you are NOT guaranteed that A = B.

The first statement is deductive and is guaranteed to be true. The second statement is inductive and, therefore, even just one exception will render the argument non-cogent (in this case, the exception where C = 0).

Answer 3

if C=0 ,certainly notbut otherwise

Answer 4

if A and B are matrices then it's possible for A not equal B