# Understanding Matrices Sums and Products

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1. Feb 13, 2016

### cosmos42

1. The problem statement, all variables and given/known data
Suppose that AB = AC for matrices A, B, and C.
Is it true that B must equal C? Prove the result or find a counterexample.

2. Relevant equations
Properties of matrix multiplication

3. The attempt at a solution
AC = A(D + B) = AD + AB = 0 + AB = AB ??? Can someone help me understand in plain english?

2. Feb 13, 2016

### Ray Vickson

If you write AB = AC as A(B-C) = 0, is it true that B-C = 0? That is, does having AD = 0 imply that D must = 0?

Last edited: Feb 14, 2016
3. Feb 14, 2016

### HallsofIvy

IF A has an inverse the AB= AC gives $A^{-1}AB= A^{-1}AC$ so $IB= IC$ so $B= C$ so you question becomes "does every matrix have an inverse?".

By the way, your question, as stated, is trivially false even for numbers: 0B= 0C for any B and C, it does NOT follow that B= C. To make you question at all interesting you should add "A non-zero".

4. Feb 14, 2016