A and B are two matrices such that $(A+B)^3=A^3+3A^2B+3AB^2+B^3$ then $ AB=BA$

Question

Let $A$ and $B$ be two invertible matrices in $M_2(\mathbb{R})$such that $(A+B)^3=A^3+3A^2B+3AB^2+B^3$ then prove or disprove that $ AB=BA$

My working:

$$(A+B)^3=A^3+3A^2B+3AB^2+B^3$$ $$\implies BA^2+B^2A+ABA+BAB =2A^2B+2AB^2$$

Now what should I do?

Answer 1

Counter-example :

$$ A= \left(\begin{array}{cc} 2 & 0 \\ & \\ 0 & 3 \\ \end{array}\right), B= \left(\begin{array}{cc} 1 & 0 \\ & \\ 1 & -10 \\ \end{array}\right). $$