The discussion revolves around the properties of matrix multiplication, specifically examining the equation AB = AC and whether it implies that B must equal C. Participants are exploring the implications of this equation under various conditions, including the existence of an inverse for matrix A.
The discussion is active, with participants offering different perspectives on the original question. Some guidance has been provided regarding the conditions under which the statement may or may not hold true, particularly concerning the existence of an inverse for matrix A.
There is an ongoing debate about the assumptions necessary for the original statement to be valid, including the implications of zero matrices and the requirement for A to be non-zero.
cosmos42 said: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?
Not helpfulHallsofIvy said: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".