Let $A$  and  $B$ are square matrices of order 2 with real elements such that  $AB=A^2{B}^2-{\left(AB\right)}^2$ and  $\det \left(B\right)=2.$  Then which of the following is/are true :

 $$\begin{gathered} AB=A^2{B}^2-{\left(AB\right)}^2 \\ AB+{\left(AB\right)}^2=A^2{B}^2 \\ \left(AB\right)(I+AB)=A^2{B}^2\Rightarrow \left(A\right)=0 \\ AB=A^2{B}^2-{\left(AB\right)}^2 \\ A=A^{2}B-ABA \\ A(I-AB+BA)=0 \end{gathered}$$
$A=0$ (or) $I=AB-BA$  (Not possible)
B)  $\det (A+2B)-\det (B+2A)$
$$\begin{gathered} =\det \left(2B\right)-\det \left(B\right) \\ =4\left(2\right)-2=6 \end{gathered}$$
C)  $tr(AB-BA)=tr\left(0\right)=0$
D)  $Det(A+B+C)=Det(B+C)$