Question

Show $\left|\begin{array}{ccc}\beta \gamma & \beta {\gamma }^{\prime }+{\beta }^{\prime }\gamma & {\beta }^{\prime }{\gamma }^{\prime }\\ \gamma \alpha & \gamma {\alpha }^{\prime }+{\gamma }^{\prime }\alpha & {\gamma }^{\prime }{\alpha }^{\prime }\\ \alpha \beta & \alpha {\beta }^{\prime }+{\alpha }^{\prime }{\beta }^{\prime }& {\alpha }^{\prime }{\beta }^{\prime }\end{array}\right|=(\alpha {\beta }^{\prime }+{\alpha }^{\prime }\beta )(\beta {\gamma }^{\prime }-{\beta }^{\prime }\gamma )(\gamma {\alpha }^{\prime }+{\gamma }^{\prime }\alpha )$.

Answer 1

Let $\Delta =\left|\begin{array}{ccc}\beta \gamma & \beta {\gamma }^{\prime }+{\beta }^{\prime }\gamma & {\beta }^{\prime }{\gamma }^{\prime }\\ \gamma \alpha & \gamma {\alpha }^{\prime }+{\gamma }^{\prime }\alpha & {\gamma }^{\prime }{\alpha }^{\prime }\\ \alpha \beta & \alpha {\beta }^{\prime }+{\alpha }^{\prime }{\beta }^{\prime }& {\alpha }^{\prime }{\beta }^{\prime }\end{array}\right|$

Taking ${\beta }^{\prime }{\gamma }^{\prime },{\gamma }^{\prime }{\alpha }^{\prime },{\alpha }^{\prime }{\beta }^{\prime }$ commons from $R_1,R_2$ and $R_3$ respectively, then
$\Delta =\left({\beta }^{\prime }{\gamma }^{\prime }\right)\left({\gamma }^{\prime }{\alpha }^{\prime }\right)\left({\alpha }^{\prime }\beta \right)\left|\begin{array}{ccc}\frac{\beta }{{\beta }^{\prime }}\frac{\gamma }{{\gamma }^{\prime }}& \frac{\beta }{{\beta }^{\prime }}+\frac{\gamma }{{\gamma }^{\prime }}& 1\\ \frac{\gamma }{{\gamma }^{\prime }}\frac{\alpha }{{\alpha }^{\prime }}& \frac{\gamma }{{\gamma }^{\prime }}+\frac{\alpha }{{\alpha }^{\prime }}& 1\\ \frac{\alpha }{{\alpha }^{\prime }}\frac{\beta }{{\beta }^{\prime }}& \frac{\alpha }{{\alpha }^{\prime }}+\frac{\beta }{{\beta }^{\prime }}& 1\end{array}\right|$

Applying $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$
then
$\Delta =({\alpha }^{\prime }{\beta }^{\prime }{\gamma }^{\prime }{)}^2\left|\begin{array}{ccc}\frac{\beta }{{\beta }^{\prime }}\frac{\gamma }{{\gamma }^{\prime }}& \frac{\beta }{{\beta }^{\prime }}+\frac{\gamma }{{\gamma }^{\prime }}& 1\\ \frac{\gamma }{{\gamma }^{\prime }}(\frac{\alpha }{{\alpha }^{\prime }}-\frac{\beta }{{\beta }^{\prime }})& (\frac{\alpha }{{\alpha }^{\prime }}-\frac{\beta }{{\beta }^{\prime }})& 0\\ \frac{\beta }{{\beta }^{\prime }}(\frac{\alpha }{{\alpha }^{\prime }}-\frac{\gamma }{{\gamma }^{\prime }})& (\frac{\alpha }{{\alpha }^{\prime }}-\frac{\gamma }{{\gamma }^{\prime }})& 0\end{array}\right|$
$=({\alpha }^{\prime }{\beta }^{\prime }{\gamma }^{\prime }{)}^2(\frac{\alpha }{{\alpha }^{\prime }}-\frac{\beta }{{\beta }^{\prime }})(\frac{\alpha }{{\alpha }^{\prime }}-\frac{\gamma }{{\gamma }^{\prime }})\left|\begin{array}{ccc}\frac{\beta }{{\beta }^{\prime }}\frac{\gamma }{{\gamma }^{\prime }}& \frac{\beta }{{\beta }^{\prime }}+\frac{\gamma }{{\gamma }^{\prime }}& 1\\ \frac{\gamma }{{\gamma }^{\prime }}& 1& 0\\ \frac{\beta }{{\beta }^{\prime }}& 1& 0\end{array}\right|$Expanding along $C_3$
$\therefore \Delta =({\alpha }^{\prime }{\beta }^{\prime }{\gamma }^{\prime }{)}^2\frac{(\alpha {\beta }^{\prime }-{\alpha }^{\prime }\beta )}{{\alpha }^{\prime }{\beta }^{\prime }}-\frac{(\alpha {\gamma }^{\prime }-{\alpha }^{\prime }\gamma )}{{\alpha }^{\prime }{\gamma }^{\prime }}(\frac{\gamma }{{\gamma }^{\prime }}-\frac{\beta }{{\beta }^{\prime }})$
$=\left({\alpha }^{\prime }{\beta }^{\prime }{\gamma }^{\prime }\right)\frac{(\alpha {\beta }^{\prime }-{\alpha }^{\prime }\beta )(\alpha {\gamma }^{\prime }-{\alpha }^{\prime }\gamma )(\gamma {\beta }^{\prime }-{\gamma }^{\prime }\beta )}{({\alpha }^{\prime }{\beta }^{\prime }{\gamma }^{\prime }{)}^2}$

$\Delta =(\alpha {\beta }^{\prime }-{\alpha }^{\prime }\beta )(\beta {\gamma }^{\prime }-{\beta }^{\prime }\gamma )(\gamma {\alpha }^{\prime }-{\gamma }^{\prime }\alpha )$

Hence Proved