Question

$\left|\begin{array}{ccc}{b}^2+c^2& a^2& a^2\\ b^2& c^2+a^2& b^2\\ c^2& c^2& a^2+b^2\end{array}\right|=$

• A. $abc$
• B. $4abc$
• C. $4a^2{b}^2{c}^2$
• D. $a^2{b}^2{c}^2$

Answer 1

The correct option is C $4a^2{b}^2{c}^2$

$\Delta =\left|\begin{array}{ccc}{b}^2+c^2& a^2& a^2\\ b^2& c^2+a^2& b^2\\ c^2& c^2& a^2+b^2\end{array}\right|=-2\left|\begin{array}{ccc}0& c^2& b^2\\ b^2& c^2+a^2& b^2\\ c^2& c^2& a^2+b^2\end{array}\right|,by{R}_1\to R_1-(R_2+R_3)=-2\left|\begin{array}{ccc}0& c^2& b^2\\ b^2& a^2& 0\\ c^2& 0& a^2\end{array}\right|,by{R}_2\to R_2-R_1{R}_3\to R_3-R_1=-2\{-c^2(b^2{a}^2)+b^2(-c^2{a}^2\left)\right\}=4a^2{b}^2{c}^2$.
Trick : Put a=1, b=2, c=3, so that the option give different values.