If $\mathrm{\Delta }\left(x\right)=\left|\begin{array}{ccc}1& x& x+1\\ 2x& x(x-1)& x(x+1)\\ 3x(x-1)& x(x-1)(x-2)& x\left(x^2-1\right)\end{array}\right|$ then $∆\left(100\right)$equals

 

Taking x common from $C_2, x + 1$ from $C_3$and x – 1 from $R_3$, we get

$\mathrm{\Delta }\left(x\right)=x(x+1)(x-1)\left|\begin{array}{ccc}1& 1& 1\\ 2x& x-1& x\\ 3x& x-2& x\end{array}\right|$

Applying $C_1\to C_1-C_3,C_2\to C_2-C_3$ we get 

$\mathrm{\Delta }\left(x\right)=x(x+1)(x-1)\left|\begin{array}{ccc}0& 0& 1\\ x& -1& x\\ 2x& 2& x\end{array}\right|=0$

                                  [ $C_1 and C_2$ are proportional]

Thus  ,         $∆ \left(100\right) =0$