Question

a,b,c are different real numbers. factor of the determinant

$\left|\begin{array}{ccc}0& x^2-a& x^3-b\\ x^2+a& 0& x^2+c\\ x^4+b& x-c& 0\end{array}\right|$ is

• A. x
• B. x-a
• C. x-b
• D. x-c

Answer 1

The correct option is A x

Here if we closely observe that the a,b and c are in opposite signs on opposite sides of the diagonal.

So if we put x=0 the determinant becomes something like this

$\left|\begin{array}{ccc}0& -a& -b\\ a& 0& c\\ b& -c& 0\end{array}\right|$, which is the determinant of a skew symmetric matrix. We know that the determinant of a skew symmetric determinant is 0.

So if we put x=0 in the determinant then the value of determinant becomes zero. So x-0 is a factor of the above mentioned determinant which is the correct answer.