Question

$a^{-1}+b^{-1}+c^{-1}=0$ such that $\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=△$ then the value of $△$ is

• A. $0$
• B. $abc$
• C. $-abc$
• D. $Noneofthese$

Answer 1

Answer: (B)

$abc$

$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=△R_2\to R_2-R_3{R}_3\to R_3-R_1\Rightarrow \left|\begin{array}{ccc}1+a& 1& 1\\ 0& b& -c\\ -a& 0& c\end{array}\right|=△\Rightarrow (1+a)\left(bc\right)-a(-c-b)=△\Rightarrow (1+a)bc+a(b+c)=△\Rightarrow abc[\frac{(1+a)bc}{abc}+\frac{a(b+c)}{abc}]=△\Rightarrow abc[\frac{1}{a}+1+\frac{1}{c}+\frac{1}{b}]=△\Rightarrow abc[1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}]=△\Rightarrow abc[1+0]=△\Rightarrow abc=△$