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  • If x, y, z are all different from zero and , then value of x-1 + y-1 + z-1 is A. x y z

If x, y, z are all different from zero and ,\left|\begin{array}{ccc} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{array}\right|=0, then value of x^{-1} + y^{-1} + z^{-1} is

\\A. x y z\\B. x^{-1} y^{-1} z^{-1}\\C. -x -y -z\\D. -1

Answers (1)

We have

\left|\begin{array}{ccc} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{array}\right|=0

\text { Apply } \mathrm{C}_{1} \rightarrow \mathrm{C}_{1}-\mathrm{C}_{3} \text { and } \mathrm{C}_{2} \rightarrow \mathrm{C}_{2}-\mathrm{C}_{3}

\left|\begin{array}{ccc} x & 0 & 1 \\ 0 & y & 1 \\ -z & -z & 1+z \end{array}\right|=0

Expand along Row 1
\\ \vspace{\baselineskip} \Rightarrow x [y (1 + z) + z] -0 + 1 (yz) = 0\\ xy + xyz + xz + yz = 0\\ \vspace{\baselineskip}

Divide both side by xyz

\\ \Rightarrow \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+1=0 \\ \therefore \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=x^{-1}+y^{-1}+z^{-1}=-1

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