If $A =\frac{1}{2}\left[\begin{array}{cc}1 & \sqrt{3} \\ -\sqrt{3} & 1\end{array}\right]$, then :

If $A$ and $B$ are square matrices of same order which $AB = A$, $BA = B$, then $(A + I)^5$ is equal to (where $I$ is unit matrix)

If $A$ is nilpotent matrix of index $2$, then $A(I_2+A)^{51}$ is equal to (where $I_2$ is the identity matrix of order $2$)

If $A = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right],B = \left[ {\begin{array}{*{20}{c}}0&{ – i}\\i&0\end{array}} \right]$ then ${(A + B)^2}$ equals

Let $A$ and $B$ be real matrices of the form $\left[ {\begin{array}{*{20}{c}}
\alpha &0\\
0&\beta 
\end{array}} \right]$ and $\left[ {\begin{array}{*{20}{c}}
0&\gamma \\
\delta &0
\end{array}} \right]$, respectively

Statement $1$ : $AB – BA$ is always an invertible matrix

Statement $2$ : $AB -BA$ is never an identity matrix