Understanding Matrix Multiplication Non-Commutativity

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Homework Help Overview

The discussion revolves around the properties of matrix multiplication, specifically addressing the non-commutativity of matrices and its implications on algebraic expressions involving matrices, such as the expansion of (A+B)^2.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the validity of the expression (A+B)^2 = A^2 + 2AB + B^2 in the context of matrices, questioning the assumptions behind the distributive law and the nature of matrix multiplication.

Discussion Status

There is an active exploration of the differences between matrix multiplication and traditional algebra. Some participants express confusion regarding the implications of non-commutativity, while others clarify the mathematical reasoning behind the statements in the study guide.

Contextual Notes

Participants note the urgency of the discussion due to an upcoming test and the need for clarity on the topic as outlined in the study guide.

PsychonautQQ
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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

 
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Hi PsychonautQQ! :wink:

In any algebra, you can use the distributive law.

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

Sooo … ? :smile:
 
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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? O_o
conflicted ;-/
 
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 [tex]A^2+ B^2+ AB+ AB= A^2+ B^2+ 2AB[/tex].
 
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