Question

If $f\left(x\right)=\left|\begin{array}{ccc}0& x-a& x-b\\ x+a& 0& x-c\\ x+b& x+c& 0\end{array}\right|$,

which of the following is true?

(a) f(a) = 0
(b) f(b) = 0
(c) f(0) = 0
(d) f(1) = 0

Answer 1

(c) We have,
$f\left(x\right)=\left|\begin{array}{ccc}0& x-a& x-b\\ x+a& 0& x-c\\ x+b& x+c& 0\end{array}\right|\Rightarrow f\left(a\right)=\left|\begin{array}{ccc}0& 0& a-b\\ 2a& 0& a-c\\ a+b& a+c& 0\end{array}\right|=\left[\right(a-b\left)\right\{2a.(a+c)\left\}\right]\ne 0\therefore f\left(b\right)=\left|\begin{array}{ccc}0& b-a& 0\\ b+a& 0& b-c\\ 2b& b+c& 0\end{array}\right|=-(b-a)\left[2b\right(b-c\left)\right]=-2b(b-a)(b-c)\ne 0\therefore f\left(0\right)=\left|\begin{array}{ccc}0& -a& -b\\ a& 0& -c\\ b& c& 0\end{array}\right|=a\left(bc\right)-b\left(ac\right)=abc-abc=0$