Understanding Matrix Multiplication Non-Commutativity

[PsychonautQQ]

Homework Statement

This isn't homework, but I didn't know where else to ask and I have a test in an hour and a half. The study guide says be able to explain why statements like (A+B)^2 = A^2 + 2AB + B^2 are bogus when dealing with matrix's. Is it because (A+B) = (B+A) but (A+B)^2 /= (B+A)^2 or something? Thx in advance

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

Hi PsychonautQQ!

In any algebra, you can use the distributive law.

So (A + B)(A + B) = AA + BB + AB + BA.

Sooo … ? ​

 

[PsychonautQQ]

well in matrix multiplaction, AB /= BA.. so in that regard it differs from normal algebra. My study guide says "Be able to explain why statements like (A+B)^2 = A^2 + 2AB + B^2 are nonsense.. and you are saying it's not nonsense?
conflicted ;-/

 

[HallsofIvy]

The crucial point is that matrix multiplication is NOT commutative. That is, in general, AB is NOT equal to BA so in tiny-tim's calculation, (A+ B)(A+ B)= AA+ BB+ AB+ BA is NOT equal to $$A^2+ B^2+ AB+ AB= A^2+ B^2+ 2AB$$.