(a) $\Delta=xyz$ $\,\left| \,\begin{matrix}  1+\frac{1}{x} & \frac{1}{x} & \frac{1}{x}  \\   \frac{1}{y} & 1+\frac{1}{y} & \frac{1}{y}  \\   \frac{1}{z} & \frac{1}{z} & 1+\frac{1}{z}  \\\end{matrix}\, \right|$

= $xyz\left( {1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right)$ $\,\left| {\,\begin{array}{*{20}{c}}1&1&1\\{\frac{1}{y}}&{1 + \frac{1}{y}}&{\frac{1}{y}}\\{\frac{1}{z}}&{\frac{1}{z}}&{1 + \frac{1}{z}}\end{array}\,} \right|$,

                                                                by ${R_1} \to {R_1} + {R_2} + {R_3}$

=$xyz\left( {1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right)$$\,\left| {\,\begin{array}{*{20}{c}}1&0&0\\{1/y}&1&0\\{1/z}&0&1\end{array}\,} \right|$, by $\begin{array}{l}{C_2} \to {C_2} – {C_1}\\{C_3} \to {C_3} – {C_1}\end{array}$

= $xyz\left( {1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right)$ $\left| {\,\begin{array}{*{20}{c}}1&0\\0&1\end{array}\,} \right| = xyz\left( {1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right)$.

Trick: Put $x = 1,\,y = 2$ and $z = 3$, then

$\left| {\,\begin{array}{*{20}{c}}2&1&1\\1&3&1\\1&1&4\end{array}\,} \right| = 2(11) – 1(3) + 1(1 – 3) = 17$

Option $(a) $ gives, $1 \times 2 \times 3\,\left( {1 + \frac{1}{1} + \frac{1}{2} + \frac{1}{3}} \right) = 17$.