$\left(\mathrm{A}\right)$
The non-singular matrix of order $3,\left|A\right|=a,$
${\left(\mathrm{adj}{A}^{\mathit{-}\mathit{1}}\right)}^{-1}\mid =\frac{1}{\left|\mathrm{adj}{A}^{\mathit{-}\mathit{1}}\right|}$

$=\frac{1}{{\left|\mathit{ }{A}^{\mathit{-}\mathit{1}}\right|}^2}$           $\left\{\left|\mathrm{adj}A\right|=|A{|}^2,\right.$ if order of $A$ is $\left.3\right\}$

$=\frac{1}{{\left(\frac{1}{| A|}\right)}^2}$

$=|A{|}^2$

$\mathit{=}{a}^{\mathit{2}}$

$\left(\mathrm{B}\right)$
It is given that for non-singular matrix $A$ of order $B\mathit{,}\mathit{ }AB\mathit{=}0$

$\left|AB\right|=0$

$\Rightarrow \left|A\right|\left|B\right| =0$

$\Rightarrow \left|B\right|=0$          $\left(\because \left|A\right|\ne 0\right)$

And matrix $B$ should be null matrix.

$\left(\mathrm{C}\right)$

$\Delta \mathit{=}\left|\begin{array}{ccc}\mathit{1}& x& x^{\mathit{2}}\\ \cos \mathit{(}a\mathit{-}b\mathit{)}y& \cos ay& \cos \mathit{(}a\mathit{+}b\mathit{)}y\\ \sin \mathit{(}a\mathit{-}b\mathit{)}y& \sin ay& \sin \mathit{(}a\mathit{+}b\mathit{)}y\end{array}\right|$

By simplifying the above equation, we get
$\sin \left(by\right)-x\sin \left(2by\right)+x^2\sin \left(by\right)$

Thus, it doesn't depend on $a.$

$\left(\mathrm{D}\right)$

It is given that

$B\mathit{=}A\mathit{-}{A}^{\mathit{T}}$

$\Rightarrow B^T\mathit{=}{\left(A\mathit{-}{A}^T\right)}^T$

$\Rightarrow B^T\mathit{=}{A}^T\mathit{-}A$

$\Rightarrow B^T\mathit{=}\mathit{-}\left(A\mathit{-}{A}^T\right)$

$\Rightarrow B^{\mathit{\top }}\mathit{=}\mathit{-}B$

$B$ is a skew symmetric matrix.
Thus, $\left|B\right|=0$