SOLUTION: Prove or disprove the following for all 2x2 matrices A and B 1) (A + B)^2 = A^2 + 2AB + B^2 2) A(A+B) = A^2 + AB I have a hard time understand proofs and its due next week.

Algebra ->  Matrices-and-determiminant -> SOLUTION: Prove or disprove the following for all 2x2 matrices A and B 1) (A + B)^2 = A^2 + 2AB + B^2 2) A(A+B) = A^2 + AB I have a hard time understand proofs and its due next week.      Log On


   

Question 807873: Prove or disprove the following for all 2x2 matrices A and B
1) (A + B)^2 = A^2 + 2AB + B^2
2) A(A+B) = A^2 + AB
I have a hard time understand proofs and its due next week.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
1) (A + B)^2 = A^2 + 2AB + B^2


(A + B)^2 = (A+B)(A+B)

(A + B)^2 = A(A+B) + B(A+B)

(A + B)^2 = A*A+A*B + B*A+B*B

(A + B)^2 = A^2+A*B + B*A+B^2

This as far as we can go. We CANNOT say that A*B = B*A since matrix multiplication is NOT commutative in general.

Therefore (A + B)^2 = A^2 + 2AB + B^2 is false.

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2) A(A+B) = A^2 + AB


A(A+B) = A*A + A*B

A(A+B) = A^2 + AB

So that confirms A(A+B) = A^2 + AB to be true.