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Question
If \[a, b\] and c are all non-zero and
Solution
We have,
\[C_1 \to C_1 - C_2 \]
\[\begin{vmatrix}a & 1 & 1 \\ - b & 1 + b & 1 \\ 0 & 1 & 1 + c\end{vmatrix} = 0\]
\[ C_2 \to C_2 - C_3 \]
\[\begin{vmatrix}a & 0 & 1 \\ - b & b & 1 \\ 0 & - c & 1 + c\end{vmatrix} = 0\]
\[\text{ Expanding along }R_1 , \text{ we get }\]
\[a(b + bc + c) + 1(bc) = 0\]
\[ \Rightarrow ab + abc + ac + bc = 0\]
\[\text{ Dividing by abc, we get }\]
\[\frac{1}{c} + 1 + \frac{1}{b} + \frac{1}{a} = 0\]
\[ \therefore \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 1 = 0\]
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