Let a,b,c are non-zero real number and $$ \Delta=\left|\begin{array}{ccc} b c & c a & a b \ c a & a

Question

Let a,b,c are non-zero real number and $$ \Delta=\left|\begin{array}{ccc} b c & c a & a b \ c a & a b & b c \ a b & b c & c a \end{array}ight| $$ where ( $\omega$ ) is cube root of unity, then $\Delta$ vanishes in which of the following cases

Professor PC Thomas Classes · Answer

$\Delta=(\mathrm{abc})^3\left|\frac{\frac{1}{\mathrm{a}}}{\frac{1}{\mathrm{~b}}} \frac{\frac{1}{\mathrm{~b}}}{\frac{1}{\mathrm{c}}} \frac{1}{\mathrm{c}}ight|=(\mathrm{abc})^3\left(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{~b}}+\frac{1}{\mathrm{c}}ight)\left|\begin{array}{ccc}1 & 1 & 1 \ \frac{1}{\mathrm{c}} & \frac{1}{\mathrm{a}} & \frac{1}{\mathrm{~b}}\end{array}ight| \frac{1}{\mathrm{~b}} \frac{1}{\mathrm{c}}\left|\frac{1}{\frac{1}{\mathrm{c}}} \frac{1}{\mathrm{c}} \frac{1}{\mathrm{~b}}ight|$ On solving $ \begin{aligned} & \Delta=\mathrm{abc}^3\left(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{~b}}+\frac{1}{\mathrm{c}}ight)\left(\frac{1}{\mathrm{bc}}+\frac{1}{\mathrm{ca}}+\frac{1}{\mathrm{ab}}-\frac{1}{\mathrm{a}^2}-\frac{1}{\mathrm{~b}^2}-\frac{1}{\mathrm{c}^2}ight) \ & \Delta=\mathrm{abc}^3\left(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{~b}}+\frac{1}{\mathrm{c}}ight)\left(\frac{1}{\mathrm{a}}+\frac{\mathrm{w}}{\mathrm{~b}}+\frac{\mathrm{w}^2}{\mathrm{c}}ight)\left(\frac{1}{\mathrm{a}}+\frac{\mathrm{w}^2}{\mathrm{~b}}+\frac{\mathrm{w}}{\mathrm{c}}ight) \end{aligned} $