In matrix multiplication, is (A-B)(A+B) = A^2-B^2?

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Answer 1 · 2015-07-09T21:01:04

No, because matrix multiplication is not commutative in general, so

$(A-B)(A+B) = A^2+AB-BA+B^2$

is not always equal to $A^2-B^2$

Since matrix multiplication is not commutative in general, take any two matrices $A$ , $B$ such that $AB != BA$ .

Then $AB-BA != 0$ so

$(A-B)(A+B) = A^2+AB-BA+B^2 != A^2+B^2$

For example, let $A=((1, 0), (0, 0))$ and $B=((0, 1), (0, 0))$

Then $AB = ((0, 1), (0, 0)) = B$ , but $BA = ((0, 0), (0, 0))$

$(A-B)(A+B) = ((1, -1), (0, 0))((1, 1), (0, 0)) =((1, 1), (0, 0))$

$A^2-B^2 = ((1, 0), (0, 0)) - ((0, 0), (0, 0)) = ((1, 0), (0, 0))$