How to prove that $ B = A^{-1} $ ? If A and B are 2 x 2 matrices where $ B \neq I_2 $ such that $ ( A + B )^2 = A^2 + 2AB + B^2 $

Question

If $A \text{ and } B \text{ are } 2\times 2$ matrices where $B\neq I_2$, such that $(A+B)^2 = A^2 +2AB + B^2$, deduce that $B = A^{-1}$

If $A = \begin{bmatrix}1&2\\9&-1\end{bmatrix}$ find $B$

.

Is there any properties involved in the solving? I don't know where to start.

Answer 1

If $A,B\in M_{n}(\mathbb{R})\backslash O_{2}$, such that $$(A+B)^{2}=A^2+2AB+B^2,$$ then A comments with B, i.e $AB=BA$