Question

Let $a,b>0$ and $\Delta =\left|\begin{array}{ccc}-x& a& b\\ b& -x& a\\ a& b& -x\end{array}\right|$, then $\Delta =$

Answer 1

Given,

$\Delta =\left|\begin{array}{ccc}-x& a& b\\ b& -x& a\\ a& b& -x\end{array}\right|$

[ $C_1^{\prime }=C_1+C_2+C_3$]

or, $\Delta =\left|\begin{array}{ccc}a+b-x& a& b\\ a+b-x& -x& a\\ a+b-x& b& -x\end{array}\right|$

or, $\Delta =(a+b-x)\left|\begin{array}{ccc}1& a& b\\ 1& -x& a\\ 1& b& -x\end{array}\right|$

[ $R_2^{\prime }=R_2-R_1$ and $R_3^{\prime }=R_3-R_1$]

or, $\Delta =(a+b-x)\left|\begin{array}{ccc}1& a& b\\ 0& -x-a& a-b\\ 0& b-a& -x-b\end{array}\right|$

or, $\Delta =(a+b-x)\left\{\right(x+a\left)\right(x+b)-(a-b{)}^2\}$