Question

The value of $x$ for which the matrix $A=\left|\begin{array}{ccc}x+a& b& c\\ a& x+b& c\\ a& b& x+c\end{array}\right|$ is non singular, is

• A. $R-\{0,-(a+b+c\left)\right\}$
• B. $R-\left\{0\right\}$
• C. $R-\{-(a+b+c\left)\right\}$
• D. None of these

Answer 1

The correct option is A $R-\{0,-(a+b+c\left)\right\}$

Given A is non singular $\therefore \left|A\right|\ne 0$

Applying $C_1\to C_1+C_2+C_3$

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

Applying $C_1\to C_1+C_2+C_3$

$\Rightarrow (x+a+b+c)\left|\begin{array}{ccc}1& b& c\\ 0& x& 0\\ 1& b& x+c\end{array}\right|\ne 0$

$\therefore$ Required values of $x$ is $R-(0,-(a+b+c\left)\right)$
Hence, option 'A' is correct.