Question

The equation $\left|\begin{array}{ccc}x& a& a\\ a& x& a\\ a& x& a\end{array}\right|+\left|\begin{array}{ccc}b& b& x\\ b& x& b\\ x& b& b\end{array}\right|$ = 0 is satisfied when x equals

• A. 0
• B. a-b
• C. a+b
• D. none of these

Answer 1

The correct option is D none of these

$\left|\begin{array}{ccc}x& a& a\\ a& x& a\\ a& x& a\end{array}\right|+\left|\begin{array}{ccc}b& b& x\\ b& x& b\\ x& b& b\end{array}\right|=0$

$R_2\to R_2-R_3$

$=\left|\begin{array}{ccc}x& a& a\\ a& x& a\\ a& x& a\end{array}\right|+\left|\begin{array}{ccc}b& b& x\\ b-x& x-b& 0\\ x& b& b\end{array}\right|=0$

$=\left|\begin{array}{ccc}b& b& x\\ b-x& x-b& b\\ x& b& b\end{array}\right|=0$

$(x-b)\left|\begin{array}{ccc}b& b& x\\ -1& 1& 0\\ x& b& b\end{array}\right|$

$R_1\to R_1-R_3$

${(x-b)}^2\left|\begin{array}{ccc}-1& 0& 1\\ -1& 1& 0\\ x& b& b\end{array}\right|=0$

$x=b$

${(x-b)}^2(-b-b-x)=0$

$x=b$ or $(-2b)$

Option D