Question

If $\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=0$ then $a^{-1}+b^{-1}+c^{-1}$ equals

• A. $1$
• B. $abc$
• C. $-1$
• D. None of these

Answer 1

Answer: (C)

$-1$

$\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=0$

$=(1+a)(1+b+c+bc-1)-1(1+c-1)+1(1-1-b)$

$b+c+bc+ab+ac+abc-c-b=0$

$bc+ab+ac=-abc$

$\Rightarrow \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=-1$