Why does $(A+B)^2 = A^2+2AB+B^2$ not work for matrices?

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Answer 1 · 2018-02-19T11:18:21

Because matrix multiplication is not generally commutative.

Matrix multiplication is associative and distributive over addition, but is not generally commutative.

So:

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

In general $BA != AB$ , so this is not the same as:

$A^2+2AB+B^2$

For example:

$((1, 1), (0, 1))((0, 1),(1, 1)) = ((1, 2), (1, 1))$

$((0, 1),(1, 1))((1, 1), (0, 1)) = ((0, 1), (1, 2))$

Why does (A+B)^2 = A^2+2AB+B^2(A+B)^2 = A^2+2AB+B^2 not work for matrices?