Question Number 30783 by math1967 last updated on 25/Feb/18
$${Prove}\:{that}\begin{vmatrix}{\mathrm{3}}&{{a}+{b}+{c}}&{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }\\{{a}+{b}+{c}}&{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }&{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }\\{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }&{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }&{{a}^{\mathrm{4}} +{b}^{\mathrm{4}} +{c}^{\mathrm{4}} }\end{vmatrix} \\ $$$$=\left({a}−{b}\right)^{\mathrm{2}} \left({b}−{c}\right)^{\mathrm{2}} \left({c}−{a}\right)^{\mathrm{2}} \\ $$
Answered by math1967 last updated on 27/Feb/18
$${Find}\begin{vmatrix}{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}\\{{a}}&{{b}}&{{c}}\\{{a}^{\mathrm{2}} }&{{b}^{\mathrm{2}} }&{{c}^{\mathrm{2}} }\end{vmatrix}\begin{vmatrix}{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}\\{{a}}&{{b}}&{{c}}\\{{a}^{\mathrm{2}} }&{{b}^{\mathrm{2}} }&{{c}^{\mathrm{2}} }\end{vmatrix} \\ $$$${then}\:{expand}\:\begin{vmatrix}{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}\\{{a}}&{{b}}&{{c}}\\{{a}^{\mathrm{2}} }&{{b}^{\mathrm{2}} }&{{c}^{\mathrm{2}} }\end{vmatrix} \\ $$