Thread: Help with basic matrix proof

"Prove or give a counterexample: If A, B, C are 2x2 matrices, AB = BC, and B is invertible, then A = C."

I'm greatly lost on how to solve this problem; I've been trying for a very long time. I've played around with numbers enough to convince myself that this is indeed a proof. However, I cannot figure out how to prove it unless A,C are either the identity matrix, the zero matrix, or the inverse of B (or if B is the identity matrix). Any help is appreciated.

EDIT:
Turns out there are counterexamples. Solutions were distributed after assignment was turned in :\ .

Originally Posted by Fenex

"Prove or give a counterexample: If A, B, C are 2x2 matrices, AB = BC, and B is invertible, then A = C."

I'm greatly lost on how to solve this problem; I've been trying for a very long time. I've played around with numbers enough to convince myself that this is indeed a proof. However, I cannot figure out how to prove it unless A,C are either the identity matrix, the zero matrix, or the inverse of B (or if B is the identity matrix). Any help is appreciated.

If B is invertible then simply multiply both sides of the equation by the inverse of B.

$\displaystyle AB = BC $

$\displaystyle ABB^{-1} = BB^{-1}C $

$\displaystyle AI = IC $

Originally Posted by Mush

If B is invertible then simply multiply both sides of the equation by the inverse of B.

$\displaystyle AB = BC $

$\displaystyle ABB^{-1} = BB^{-1}C $

$\displaystyle AI = IC $

How are you able to switch the order of C and B^-1 in the second line? Since matrix multiplication isn't commutative.