Question

Prove that $\left|\begin{array}{ccc}bc& b{c}^{\prime }+b^{\prime }c& b^{\prime }{c}^{\prime }\\ ca& c{a}^{\prime }+c^{\prime }a& c^{\prime }{a}^{\prime }\\ ab& a{b}^{\prime }+a^{\prime }b& a^{\prime }{b}^{\prime }\end{array}\right|=(a{b}^{\prime }-a^{\prime }b)(bc-b^{\prime }c)(c{a}^{\prime }-c^{\prime }a)$

Answer 1

Let $\Delta =\left|\begin{array}{ccc}bc& b{c}^{\prime }+b^{\prime }c& b^{\prime }{c}^{\prime }\\ ca& c{a}^{\prime }+c^{\prime }a& c^{\prime }{a}^{\prime }\\ ab& a{b}^{\prime }+a^{\prime }b& a^{\prime }{b}^{\prime }\end{array}\right|$
Taking $\left(b^{\prime }{c}^{\prime }\right),\left(c^{\prime }{a}^{\prime }\right)$ and $\left(a^{\prime }{b}^{\prime }\right)$ common from $R_1,R_2$ and $R_3$ respectively, we get
$\Delta =\left(b^{\prime }{c}^{\prime }\right)\left(c^{\prime }{a}^{\prime }\right)\left(a^{\prime }{b}^{\prime }\right)\left|\begin{array}{ccc}\frac{bc}{b^{\prime }{c}^{\prime }}& \frac{b}{b^{\prime }}+\frac{c}{c^{\prime }}& 1\\ \frac{ca}{c^{\prime }{a}^{\prime }}& \frac{c}{c^{\prime }}+\frac{a}{a^{\prime }}& 1\\ \frac{ab}{a^{\prime }{b}^{\prime }}& \frac{a}{a^{\prime }}+\frac{b}{b^{\prime }}& 1\end{array}\right|$
Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$
$\Delta ={\left(a^{\prime }{b}^{\prime }{c}^{\prime }\right)}^2\left|\begin{array}{ccc}\frac{bc}{b^{\prime }{c}^{\prime }}& \frac{b}{b^{\prime }}+\frac{c}{c^{\prime }}& 1\\ \frac{c}{c^{\prime }}(\frac{a}{a^{\prime }}-\frac{b}{b^{\prime }})& (\frac{a}{a^{\prime }}-\frac{b}{b^{\prime }})& 0\\ \frac{b}{b^{\prime }}(\frac{a}{a^{\prime }}-\frac{c}{c^{\prime }})& \frac{a}{a^{\prime }}-\frac{c}{c^{\prime }}& 0\end{array}\right|\Delta ={\left(a^{\prime }{b}^{\prime }{c}^{\prime }\right)}^2(\frac{a}{a^{\prime }}-\frac{b}{b^{\prime }})(\frac{a}{a^{\prime }}-\frac{c}{c^{\prime }})\left|\begin{array}{ccc}\frac{bc}{b^{\prime }{c}^{\prime }}& \frac{b}{b^{\prime }}+\frac{c}{c^{\prime }}& 1\\ \frac{c}{c^{\prime }}& 1& 0\\ \frac{b}{b^{\prime }}& 1& 0\end{array}\right|\Delta ={\left(a^{\prime }{b}^{\prime }{c}^{\prime }\right)}^2\frac{a{b}^{\prime }-a^{\prime }b}{a^{\prime }{b}^{\prime }}\frac{a{c}^{\prime }-a^{\prime }c}{a^{\prime }{c}^{\prime }}(\frac{c}{c^{\prime }}-\frac{b}{b^{\prime }})=(a{b}^{\prime }-a^{\prime }b)(bc-b^{\prime }c)(c{a}^{\prime }-c^{\prime }a)$