Question

The value of sum of the given determinants,
$|A|=\left|\begin{array}{ccc}103& 115& 114\\ 111& 108& 106\\ 104& 113& 116\end{array}\right|,|B|=\left|\begin{array}{ccc}113& 116& 104\\ 108& 106& 111\\ 115& 114& 103\end{array}\right|$ is

Answer 1

In the det$\left(A\right),$ first interchange $R_1$ and $R_3$
$\Rightarrow |A|=-\left|\begin{array}{ccc}104& 113& 116\\ 111& 108& 106\\ 103& 115& 114\end{array}\right|$
Now interchange $C_1$ and $C_2$
$\Rightarrow |A|=\left|\begin{array}{ccc}113& 104& 116\\ 108& 111& 106\\ 115& 103& 114\end{array}\right|$
Now interchange $C_2$ and $C_3$
$\Rightarrow |A|=-\left|\begin{array}{ccc}113& 116& 104\\ 108& 106& 111\\ 115& 114& 103\end{array}\right|=-|B|$
$\Rightarrow |A|+|B|=0$