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If $\mathrm{a}, \mathrm{b}, \mathrm{c}$, are non-zero real numbers and
$\left[\begin{array}{ccc}1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{array}\right]=0$
then what is the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ ?
Options:
$\left[\begin{array}{ccc}1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{array}\right]=0$
then what is the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ ?
Solution:
1631 Upvotes
Verified Answer
The correct answer is:
$-1$
Given $\left|\begin{array}{ccc}1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{array}\right|=0$
Applying $C_{2} \rightarrow C_{2}-C_{1}$ and $C_{3} \rightarrow C_{3}-C_{1}$
$\Rightarrow\left|\begin{array}{ccc}1+a & -a & -a \\ 1 & b & 0 \\ 1 & 0 & c\end{array}\right|=0$
$\Rightarrow$ Expanding along $R_{3}, 1(a b)+c(b+a b+a)=0$
$\Rightarrow a b+b c+c a+a b c=0$
$\Rightarrow \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=-1$
Applying $C_{2} \rightarrow C_{2}-C_{1}$ and $C_{3} \rightarrow C_{3}-C_{1}$
$\Rightarrow\left|\begin{array}{ccc}1+a & -a & -a \\ 1 & b & 0 \\ 1 & 0 & c\end{array}\right|=0$
$\Rightarrow$ Expanding along $R_{3}, 1(a b)+c(b+a b+a)=0$
$\Rightarrow a b+b c+c a+a b c=0$
$\Rightarrow \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=-1$
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