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

Construct a $3 \times 4$ matrix, whose elements are given by $a_{i j}=\frac{1}{2}|-3 i+j|$.

If $A = \left[ {\begin{array}{*{20}{c}}{{{\cos }^2}\theta }&{\sin \theta \cos \theta }\\{\sin \theta \cos \theta }&{{{\sin }^2}\theta }\end{array}} \right]$, $B = \left[ {\begin{array}{*{20}{c}}{{{\cos }^2}\phi }&{\sin \phi \cos \phi }\\{\sin \phi \cos \phi }&{{{\sin }^2}\phi }\end{array}} \right]$ and $\theta $ and $\phi $ differs by $\frac{\pi }{2},$ then $AB = $

$AB = 0$, if and only if

$\left[ {\begin{array}{*{20}{c}}7&1&2\\9&2&1\end{array}} \right]\,\left[ \begin{array}{l}3\\4\\5\end{array} \right] + 2\left[ \begin{array}{l}4\\2\end{array} \right]$ is equal to