Question

The divisors of the determinant $D=\begin{vmatrix}\frac{a^2+b^2}{c} & c & c \\ a & \frac{b^2+c^2}{a} & a \\ b & b & \frac{c^2+a^2}{b}\end{vmatrix}$ 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)

$\frac{1}{a b c}\begin{vmatrix}a^2+b^2 & c^2 & c^2 \\ a^2 & b^2+c^2 & a^2 \\ b^2 & b^2 & c^2+a^2\end{vmatrix}$
$\text { use } R_1 \rightarrow R_1-\left(R_2+R_3\right) $
$\frac{1}{a b c}\begin{vmatrix}0 & -2 b^2 & -2 a^2 \\a^2 & b^2+c^2 & a^2 \\b^2 & b^2 & c^2+a^2\end{vmatrix}$
$R_2 \rightarrow R_2+1 / 2 R_1 \text { and } R_3 \rightarrow R_3+1 / 2 R_1$
$\frac{1}{a b c}\begin{vmatrix}0 & -2 b^2 & -2 a^2 \\a^2 & c^2 & 0 \\b^2 & 0 & c^2\end{vmatrix}$
$\frac{1}{a b c}\left[2 b^2\left(a^2 c^2\right)-2 a^2\left(-b^2 c^2\right)\right]=\frac{4 a^2 b^2 c^2}{a b c}=4 a b c$