Understanding Matrices Sums and Products

[cosmos42]

Homework Statement

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.

Homework Equations

Properties of matrix multiplication

The Attempt at a Solution

AC = A(D + B) = AD + AB = 0 + AB = AB ? Can someone help me understand in plain english?

 

[Ray Vickson]

Homework Statement

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.

Homework Equations

Properties of matrix multiplication

The Attempt at a Solution

AC = A(D + B) = AD + AB = 0 + AB = AB ? Can someone help me understand in plain english?

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?

 

[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".

 

[cosmos42]

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".

Not helpful