Question

The divisors of the determinant $D=\left|\begin{array}{ccc}\frac{a^2+b^2}{c}& c& c\\ a& \frac{b^2+c^2}{a}& a\\ b& b& \frac{c^2+a^2}{b}\end{array}\right|$ is $(a,b,c\in R)$

• A. abc
• B. $(a-b)(b-c)(c-a)$
• C. $a^3+b^3{c}^3$
• D. $\left(a^2+b^2+c^2\right)(a+b+c)$

Answer 1

Answer: (A)

$\frac{1}{abc}\left|\begin{array}{ccc}{a}^2+b^2& c^2& c^2\\ a^2& b^2+c^2& a^2\\ b^2& b^2& c^2+a^2\end{array}\right|$
$\text{ use }{R}_1\to R_1-\left(R_2+R_3\right)$
$\frac{1}{abc}\left|\begin{array}{ccc}0& -2b^2& -2a^2\\ a^2& b^2+c^2& a^2\\ b^2& b^2& c^2+a^2\end{array}\right|$
$R_2\to R_2+1/2R_1\text{ and }{R}_3\to R_3+1/2R_1$
$\frac{1}{abc}\left|\begin{array}{ccc}0& -2b^2& -2a^2\\ a^2& c^2& 0\\ b^2& 0& c^2\end{array}\right|$
$\frac{1}{abc}\left[2b^2\left(a^2{c}^2\right)-2a^2\left(-b^2{c}^2\right)\right]=\frac{4a^2{b}^2{c}^2}{abc}=4abc$