Question

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

Answer 1

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`

Explanation:

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))`