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))#