Question

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

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

Answer 1

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

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

$R_1\to R_1-R_2$

$\left[\begin{array}{ccc}{c}^1& -c^2& a^2-b^2\\ b^2& c^2+a^2& b^2\\ c^2& c^2& a^2+b^2\end{array}\right]$

$R_1\to R_1+R_3$

$\left[\begin{array}{ccc}2c^2& 0& 2a^2\\ b^2& c^2+a^2& b^2\\ c^2& c^2& a^2+b^2\end{array}\right]$

$R_3\to R_3-R_2$

$2\left[\begin{array}{ccc}{c}^2& 0& a^2\\ b^2& c^2+a^2& b^2\\ c^2-b^2& -a^2& a^2\end{array}\right]$

$R_3\to R_3-R_1$

$detA=2\left[\begin{array}{ccc}{c}^2& 0& a^2\\ b^2& c^2+a^2& b^2\\ -b^2& -a^2& 0\end{array}\right]$

$detA=2\left[c^2\right[(c^2+a^2)\left(0\right)-(-a^2)\left(b^2\right)]+a^2[-a^2{b}^2-\left(b^2\right)-(-b^2)(c^2+a^2)]$

$\Rightarrow 2[a^2{c}^2{b}^2-a^4{b}^2+a^2{b}^2{c}^2+a^4{b}^2]$

$\Rightarrow 4a^2{b}^2{c}^2$