If,x,y and z, and are all different from zero and $$1$$ $$+$$ x $$1$$ $$1$$ $$1$$ $$1$$ $$+$$ y $$1$$ $$1$$ $$1$$ $$1$$ $$+$$ $$z=0$$, then the value of x−$$1+y$$−$$1+z$$−$$1is$$

Question

If,x,y and z, and are all different from zero and $$1$$ $$+$$ x    $$1$$       $$1$$   $$1$$     $$1$$ $$+$$ y    $$1$$   $$1$$       $$1$$     $$1$$ $$+$$ $$z=0$$, then the value of x−$$1+y$$−$$1+z$$−$$1is$$

Infinity Learn · Accepted Answer

we have,$$1$$ $$+$$ x    $$1$$       $$1$$   $$1$$     $$1$$ $$+$$ y    $$1$$   $$1$$       $$1$$     $$1$$ $$+$$ $$z=0applying$$ $$C1$$→$$C1$$−$$C3$$ and $$C2$$→$$C2$$−$$C3$$⇒ x   $$0$$       $$1$$ $$0$$    y      $$1$$ $$-z$$  $$-z$$    $$1+z=0$$   Expanding along $$R1$$⇒x[$$y(1+z)+z$$]$$+0+1(yz)=0$$⇒$$x(y+yz+z)+yz=0$$⇒$$xy+xyz+xz+yz=0$$⇒$$xyxyz+xyzxyz+xzxyz+yzxyz=0$$                                                                on dividing (xyz) from both sides⇒$$1x+1y+1z+1=0$$⇒$$1x+1y+1z=$$−$$1$$∴x−$$1+y$$−$$1+z$$−$$1=$$−$$1$$

If, $x, y a n d z,$ and are all different from zero and $|\begin{array}{l} 1 + x 1 1 \\ 1 1 + y 1 \\ 1 1 1 + z \end{array}| = 0$ , then the value of $x^{- 1} + y^{- 1} + z^{- 1}$ is

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If,x,y and z, and are all different from zero and 1 + x 1 1 1 1 + y 1 1 1 1 + z=0, then the value of x−1+y−1+z−1is

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If, $x, y a n d z,$ and are all different from zero and $|\begin{array}{l} 1 + x 1 1 \\ 1 1 + y 1 \\ 1 1 1 + z \end{array}| = 0$ , then the value of $x^{- 1} + y^{- 1} + z^{- 1}$ is