Question

If $a,b,c$ are different real numbers and $x$ is a variable then a 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. $x+a$

Answer 1

The correct option is A

$x$

Here if we closely observe that $a,b\text{and}c$ 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 matrix 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.