Need help to find my mistake in a simple proof of a matrix algebra proposition.

Brutus

1. The problem statement, all variables and given/known data
Is the following true for matrices?

Hypotesis:
AB = AC
A != 0(zero matrix)

Thesis:
B=C

3. The attempt at a solution

AB = AC
AB - AC = 0(zero matrix)
AB - AC = A(B-C) // using the following property: A(B+C) = AB + AC iff A is mn matrix and BC are np matrices
A(B-C) = 0 <=> B=C because A != 0
QED

There is something wrong because there are matrices where AB = AC and B != C.
Where is my mistake?

Dick

There are matrices where AB=0 and neither A nor B are zero. You can't say A(B-C)=0 implies A=0 or (B-C)=0 like you can with real numbers.

Brutus

ok, thanks