Question

If $\Delta =\left|\begin{array}{ccc}-x& a& b\\ b& -x& a\\ a& b& -x\end{array}\right|$, then a factor of $\Delta$ is

• A. $a+b+x$
• B. $a-b-x$
• C. $a-b+x$
• D. $a+b-x$

Answer 1

The correct option is D $a+b-x$

Using $C_1\to C_1+C_2+C_3$, we get
$\Delta =\left|\begin{array}{ccc}a+b-x& a& b\\ a+b-x& -x& a\\ a+b-x& b& -x\end{array}\right|=(a+b-x)\left|\begin{array}{ccc}1& a& b\\ 1& -x& a\\ 1& b& -x\end{array}\right|=(a+b-x)(x^2+(a+b)x+a^2+b^2-ab)$