Mathematics - Determinant Question with Solution | TestHub

$$\begin{array}{rl}& \left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+(-1{)}^n\left|\begin{array}{ccc}a+1& b+1& c-1\\ a-1& b-1& c+1\\ a& -b& c\end{array}\right|=\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+(-1{)}^n\left|\begin{array}{ccc}a+1& a-1& a\\ b+1& b-1& -b\\ c-1& c+1& c\end{array}\right|\\ & =\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+(-1{)}^{n+1}\left|\begin{array}{ccc}a+1& a& a-1\\ b+1& -b& b-1\\ c-1& c& c+1\end{array}\right|=\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+(-1{)}^{n+2}\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|\end{array}$$ This is equal to zero only if $n+2$ is odd i.e. $n$ is odd integer.