Question

If $\Delta=\begin{vmatrix}-x&a&b\\ b&-x&a\\ a&b&-x\end{vmatrix}$, then a factor of $\Delta$ is

• A. $a+b+x$
• B. $x^{2}-(a-b) x+a^{2}+b^{2}+ab$
• C. $x^{2}+(a+b) x+a^{2}+b^{2}-ab$
• D. $x$

Answer 1

Answer: (C)

Applying $C_{1} \rightarrow C_{1}+C_{2}+C_{3}$ we get
$\Delta= \begin{vmatrix}a+b-x&a&b\\ a+b-x&-x&b\\ a+b-x&b&-x\end{vmatrix} =\left(a+b-x\right) \begin{vmatrix}1&a&b\\ 1&-x&a\\ 1&b&-x\end{vmatrix}$
$=\left(a+b-x\right) \begin{vmatrix}1&a&b\\ 0&-x-a&a-b\\ 0&b-a&-x-b\end{vmatrix} $
[using $R_{2} \rightarrow R_{2}-R_{1}$ and
$\left.R_{3} \rightarrow R_{3}-R_{1}\right]$
$=(a+b-x)\left[(x+a)(x+b)+(a-b)^{2}\right] $ [expanding along $C_{1}]$
$=(a+b-x)\left[x^{2}+(a+b) x+a^{2}+b^{2}-a b\right]$