What the Binomial Formula Leaves Behind

$A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

For ordinary numbers, $(a+b)^2 = a^2 + 2ab + b^2$, so the expression $(a+b)^2 - a^2 - 2ab - b^2$ is always $0$. Matrices are not ordinary numbers.

Evaluate

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

where $(A+B)^2$ means the matrix product $(A+B)(A+B)$, and $2AB$ means the matrix $AB$ with every entry doubled.