Question

The determinant $\Delta =\left|\begin{array}{ccc}{a}^2+x^2& ab& ac\\ ab& b^2+x^2& bc\\ ac& bc& c^2+x^2\end{array}\right|$ is divisible by

• A. x
• B. $x^2$
• C. $x^3$
• D. $x^4$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A x
B $x^2$
C $x^4$
D $x^3$

$\Delta =\left|\begin{array}{ccc}{a}^2+x^2& ab& ac\\ ab& b^2+x^2& bc\\ ac& bc& c^2+x^2\end{array}\right|$

Expanding it,

$=(a^2+x^2)\left(x^2\right)(b^2+c^2+x^2)-ab\left(ab{x}^2\right)+ac(-ac{x}^2)$

$=x^2(a^2{b}^2+a^2{c}^2+a^2{x}^2+x^2{b}^2+x^2{c}^2+x^4-a^2{b}^2-a^2{c}^2)$

$=x^2(a^2{x}^2+x^2{b}^2+x^2{c}^2+x^4)$

$=x^4(a^2+b^2+c^2+x^2)$

So it is divisible by $x,x^2,x^{3}and{x}^4$