Question

$\left|\begin{array}{ccc}{b}^2+c^2& ab& ac\\ ba& c^2+a^2& bc\\ ca& cb& a^2+b^2\end{array}\right|=4a^2{b}^2{c}^2$

Answer 1

$∆=\left|\begin{array}{ccc}{b}^2+c^2& ab& ac\\ ba& c^2+a^2& bc\\ ca& cb& a^2+b^2\end{array}\right|$

$\left|\begin{array}{ccc}a(b^2+c^2)& a^{2}b& a^{2}c\\ b^{2}a& b(c^2+a^2)& b^{2}c\\ c^{2}a& c^{2}b& c(a^2+b^2)\end{array}\right|\left[\mathrm{Multiplying}\mathrm{the}\mathrm{three}rows\mathrm{by}a,b\mathrm{and}c\right]$


$$\begin{gathered} =\frac{abc}{abc}\left|\begin{array}{ccc}{b}^2+c^2& a^2& a^2\\ b^2& c^2+a^2& b^2\\ c^2& c^2& a^2+b^2\end{array}\right|\left[\mathrm{Taking}\mathrm{out}a,bandc\mathrm{common}\mathrm{from}\mathrm{the}\mathrm{three}\mathrm{columns}\right] \\ =\left|\begin{array}{ccc}2(b^2+c^2)& 2(a^2+c^2)& 2(a^2+b^2)\\ b^2& c^2+a^2& b^2\\ c^2& c^2& a^2+b^2\end{array}\right|\left[\mathrm{Applying}{R}_1\to R_1+R_2+R_3\right] \\ =2\left|\begin{array}{ccc}{b}^2+c^2& a^2+c^2& a^2+b^2\\ -c^2& 0& -a^2\\ -b^2& -a^2& 0\end{array}\right|\left[\mathrm{Taking}\mathrm{out}2\mathrm{common}\mathrm{from}\mathrm{the}\mathrm{three}\mathrm{columns}\mathrm{and}\mathrm{then}\mathrm{applying}{R}_2\to R_2-R_1\mathrm{and}{R}_3\to R_3-R_1\right] \\ =2\left|\begin{array}{ccc}0& c^2& b^2\\ -c^2& 0& -a^2\\ -b^2& -a^2& 0\end{array}\right|\left[\mathrm{Applying}{R}_1\to R_1+R_2+R_3\right] \\ =2\left\{\right[-c^2(-a^2{b}^2)]+[b^2\left(c^2{a}^2\right)\left]\right\}\left[\mathrm{Expanding}\mathrm{along}{R}_1\right] \\ =4a^2{b}^2{c}^2 \end{gathered}$$