Question

If $x\ne 0,y\ne 0,z\ne 0$ and $\left|\begin{array}{ccc}1+x& 1& 1\\ 1+y& 1+2y& 1\\ 1+z& 1+z& 1+3z\end{array}\right|=0$, then $x^{-1}+y^{-1}+z^{-1}$ is equal to

• A. $-1$
• B. $-2$
• C. $-3$
• D. none of these

Answer 1

The correct option is C $-3$

$\left|\begin{array}{ccc}1+x& 1& 1\\ 1+y& 1+2y& 1\\ 1+z& 1+z& 1+3z\end{array}\right|=0$
$xyz\left|\begin{array}{ccc}\frac{1}{x}+1& \frac{1}{x}& \frac{1}{x}\\ \frac{1}{y}+1& \frac{1}{y}+2& \frac{1}{y}\\ \frac{1}{z}+1& \frac{1}{z}+1& \frac{1}{z}+3\end{array}\right|=0$
$C_1\to C_1-C_2,C_2\to C_2-C_3$
$xyz\left|\begin{array}{ccc}1& 0& \frac{1}{x}\\ -1& 2& \frac{1}{y}\\ 0& -2& \frac{1}{z}+3\end{array}\right|=0$

$\Rightarrow$$2xyz\left|\begin{array}{ccc}1& 0& \frac{1}{x}\\ -1& 1& \frac{1}{y}\\ 0& -1& \frac{1}{z}+3\end{array}\right|=0$

$2xyz[\frac{1}{z}+3+\frac{1}{y}+\frac{1}{x}]=0$
$\Rightarrow \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+3=0$
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=-3$