Sum of the roots of the equation $$ \left|\begin{array}{cccc} x & 0 & 0 & 0 \ 0 & 0 & 0 & 1 \ 0 & x & 0 & 0 \ 2 & 0 & x-1 & 0 \end{array} ight|-\left|\begin{array}{ccc} 0 & x & 0 \ 0 & 0 & x-1 \ 2 & 2 & 0 \end{array} ight|=0 \mathrm{is} $$

Question

Sum of the roots of the equation $$ \left|\begin{array}{cccc} x & 0 & 0 & 0 \ 0 & 0 & 0 & 1 \ 0 & x & 0 & 0 \ 2 & 0 & x-1 & 0 \end{array}ight|-\left|\begin{array}{ccc} 0 & x & 0 \ 0 & 0 & x-1 \ 2 & 2 & 0 \end{array}ight|=0 \mathrm{is} $$

Aicharya · Accepted Answer

$\left|\begin{array}{cccc}x & 0 & 0 & 0 \ 0 & 0 & 0 & 1 \ 0 & x & 0 & 0 \ 2 & 0 & x-1 & 0\end{array}ight|-\left|\begin{array}{ccc}0 & x & 0 \ 0 & 0 & x-1 \ 2 & 2 & 0\end{array}ight|=0$ On expanding first determinant along $R_2$ and second determinant along $R_i$ $ \begin{aligned} & 1\left[x^2(x-1)ight]-(-x)[-2(x-1)]=0 \ & x^3-x^2+x(-2 x+2)=0 \ & x^3-x^2-2 x^2+2 x=0 \ & x^3-3 x^2+2 x=0 \ & x\left(x^2-3 x+2ight)=0 \ & x(x-1)(x-2)=0 \ & x=0,1 	ext { and } 2 \ & \Rightarrow \quad \end{aligned} $ $x=0,1$ and 2 $	herefore$ Sum of the roots $=0+1+2=3$

Sum of the roots of the equation $$ \left|\begin{array}{cccc} x & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & x & 0 &…

Solution