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

1 Answer
Feb 19, 2018

Because matrix multiplication is not generally commutative.

Explanation:

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))#