If x, y, z are non-zero real numbers such that \( \begin{vmatrix}10+x & 10 & 10\\ 10 & 10+y & 10\\ 10 & 10 & 10+z\end{vmatrix}=0 \), then the value of \( x^{-1}+y^{-1}+z^{-1} \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( -\displaystyle\frac{1}{10} \)
The row operations \( R_{1}\to R_{1}-R_{2} \) and \( R_{2}\to R_{2}-R_{3} \) give
\( \displaystyle \begin{vmatrix}x&-y&0\\0&y&-z\\10&10&10+z\end{vmatrix}=0 \)
Expanding along the first row,
\( \displaystyle x\left[y(10+z)+10z\right]+y\left[10z\right]=0 \)
\( \displaystyle 10xy+xyz+10xz+10yz=0 \)
Dividing by xyz (all non-zero),
\( \displaystyle 1+10\left(\frac{1}{z}+\frac{1}{y}+\frac{1}{x}\right)=0 \)
\( \displaystyle x^{-1}+y^{-1}+z^{-1}=-\frac{1}{10} \)