Correct option: B.$-\frac{1}{165}\left[ \begin{matrix} 74 & -67 \\ -61 & 53 \\ \end{matrix} \right]$
B
$AB=\left[\begin{matrix}3 & 5\\2 & 7\\\end{matrix}\right]\left[\begin{matrix}6 & 9\\7 & 8\\\end{matrix}\right]$
$=\left[\begin{matrix}18+35 & 27+40\\12+49 & 18+56\\\end{matrix}\right]$
$\left[\begin{matrix}53 & 67\\61 & 74\\\end{matrix}\right]$
$|AB|\left[ \begin{matrix}53&67\\61&74\\\end{matrix}\right]=3922-4087=-165\neq0$
$\therefore\left(AB^{-1}\right)$નું અસિતત્વ છે.
$\therefore abjAB=\left[\begin{matrix}74 & -67\\- 61 & 53\\\end{matrix}\right]$
$\therefore \left(AB^{-1}\right)=\frac{1}{|AB|} abj(AB)$
$=-\frac{1}{165}\left[\begin{matrix}74 & -67\\- 61 & 53\\\end{matrix}\right]$ ...............................$1$
$\therefore|B|=\begin{vmatrix}6 & 9\\7 & 8\\\end{vmatrix}=48-63=-15$ 
$0$
$B^{-1}$ નું અસ્તિત્વ છે.
$\therefore abjB=\begin{vmatrix}8 & -9\\-7 & 6\\\end{vmatrix}$
$\therefore B^{-1}=\frac{1}{|B|}abjB=-\frac{1}{15}\left[\begin{matrix}8 & -9\\-7 & 6\\\end{matrix}\right]$
$|A|=\begin{vmatrix}3 & 5\\2 & 7\\\end{vmatrix}=21-10=11$ $0$
$\therefore A^{-1}$ નું અસિતત્વ છે.
$\therefore abjA =\left[\begin{matrix}7 & -5\\-2 & 3\\\end{matrix}\right]$
$\therefore A^{-1}=\frac{1}{|A|}abjA=\frac{1}{11}\left[\begin{matrix}7 & -5\\-2 & 3\\\end{matrix}\right]$
હવે,$B^{-1}\cdot A^{-1}=\frac{-1}{15}.\frac{1}{11}\left[\begin{matrix}8 & -9\\-7 & 6\\ \end{matrix} \right] \left[\begin{matrix}7 & -5\\-2 & 3\\\end{matrix}\right]$
$=\frac{-1}{165}\left[\begin{matrix}56+18 & -40-27\\-49-12 & 35+18\\\end{matrix}\right]$
$=\frac{-1}{165}\left[\begin{matrix}74 & -67\\-61 & 53\\\end{matrix}\right]$ ..................................$(2)$
$(1)$ અને $(2)$ પરથી
$\left(AB\right)^{-1}=B^{-1}A^{-1}$