Question

Prove the following identities:
$\left|\begin{array}{ccc}1& bc+ad& b^2{c}^2+a^2{d}^2\\ 1& ca+bd& c^2{a}^2+b^2{d}^2\\ 1& ab+cd& a^2{b}^2+c^2{d}^2\end{array}\right|=-(b-c)(c-a)(a-b)(a-d)(b-d)(c-d)$.

Answer 1

$\left|\begin{array}{ccc}1& bc+ad& b^2{c}^2+a^2{d}^2\\ 1& ca+bd& c^2{a}^2+b^2{d}^2\\ 1& ab+cd& a^2{b}^2+c^2{d}^2\end{array}\right|$

$=\left|\begin{array}{ccc}1& bc+ad& (bc+ad{)}^2\\ 1& ca+bd& (ca+bd{)}^2\\ 1& ab+cd& (ab+cd{)}^2\end{array}\right|[C_3\to C_3+2abcd\cdot C_1]$

$=\left|\begin{array}{ccc}1& bc+ad& (bc+ad{)}^2\\ 0& (a-b)(c-d)& (a-b)(c-d)(ac+bd+bc+ad)\\ 0& (a-c)(b-d)& (a-c)(b-d)(ab+cd+bc+ad)\end{array}\right|[R_2\to R_2–R_1,R_3\to R_3–R_1]$

$=(a-b)(c-d)(a-c)(b-d)(ab+cd+bc+ad-ac-bd–bc–ad)=-(b-c)(c-a)(a-b)(a-d)(b-d)(c-d)$