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.

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) (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.

=======================================================

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.
RELATED QUESTIONS
Prove or disprove that Segment AB ~= Segment BC A (-2,0) B (10, -14) C (-4,... (answered by LinnW)

1-2ab-(a+b)^2 (answered by It is costly)

(a^2+b^2+2ab)-(a^2+b^2-2ab) (answered by jim_thompson5910)

The following appears to prove that any two numbers are equal. Obviously it is wrong; can (answered by Edwin McCravy)

The following appears to prove that any two numbers are equal. Obviously it is wrong; can (answered by vleith)

a^2... (answered by jim_thompson5910)

If {{{a^2-2ab+b^2=36}}} and {{{a^2-3ab+b^2=22}}} find... (answered by richard1234)

[(A^3-B^3)/(A^2-b^2)] /... (answered by DrBeeee)

Multiply ( 1/a^2 - 1/b^2 ) ( 2ab / a-b) (answered by drk)