# Linear Algebra question concerning matrices

1. Oct 18, 2005

### insertnamehere

Hi, I have a question about matrices.
If A,B,C are matrices such that AB=AC and A is not equal to zero, does it follow that B=C?
I looked at the associative laws that A(BC)=(AB)C=B(AC), and I think that B and C would be the same. Am I on the right track?

2. Oct 18, 2005

### enigma

Staff Emeritus
I'd check that associative law again. The first two work, but because you can't multiply matrices together unless their ranks match up correctly, the B(AC) will not work for all cases.

For the AB=AC... you're on the right track. Try multiplying by a matrix and see if you can prove it for yourself. Another hint is below in white if you need it

You'll need to do something to A and premultiply

3. Oct 19, 2005

### shmoe

Of course you mean the numbers of rows and columns of the matrices are such that the multiplication B(AC) might not be defined even if A(BC) is. "Rank" is a bad word here as it has another meaning (I'm being picky, but thought it was worth clarifying).

Another problem with insertnamehere's associativity law is swapping the order of multiplication of matrices is not allowed in general (multiplication is not commutative).

As to AB=AC implying B=C, associativitity has nothing to do with it. Try considering the case when C=0. Then your proposition says "if A is nonzero and AB=0 then B is non zero". Is this always true? (it might help to think about 2x2 examples here)