$A = \left[ {\begin{array}{*{20}{c}}
\alpha &0\\
0&\beta 
\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}
0&\lambda \\
\delta &0
\end{array}} \right]$

Now, $AB = \left[ {\begin{array}{*{20}{c}}
0&{\alpha \gamma }\\
{\beta \delta }&0
\end{array}} \right]$

and $BA = \left[ {\begin{array}{*{20}{c}}
0&{\gamma \beta }\\
{\delta \alpha }&0
\end{array}} \right]$

Statement - $1$:

$AB - BA = \left[ {\begin{array}{*{20}{c}}
0&{\gamma \left( {\alpha  - \beta } \right)}\\
{\delta \left( {\beta  - \alpha } \right)}&0
\end{array}} \right]$

$\left| {AB - BA} \right| = {\left( {\alpha  - \beta } \right)^2}\delta  \ne 0$

$\therefore AB - BA$is always an invertible matrix.

Hence, statement - $1$ is true.

But $AB - BA$ can be identity matrix if $\gamma  =  - \delta $ or $\delta  =  - \gamma $

So, statement - -$2$ is false.