Question

The value of determinant $\Delta =\left|\begin{array}{ccc}1+a& b& c\\ a& 1+b& c\\ a& b& 1+c\end{array}\right|$ is equal to

• A. $a+b+c+1$
• B. $abc-1$
• C. $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1$
• D. $a^2+b^2+c^2+1$

Answer 1

The correct option is A $a+b+c+1$

Given : $\Delta =\left|\begin{array}{ccc}1+a& b& c\\ a& 1+b& c\\ a& b& 1+c\end{array}\right|$
Put $a=1,b=2,c=3$
determinant becomes,
$\Delta =\left|\begin{array}{ccc}2& 2& 3\\ 1& 3& 3\\ 1& 2& 4\end{array}\right|=2(12-6)-2(4-3)+3(2-3)=7$
By putting values of $a,b$ and $c$ in options.
$\therefore$ Only option $\left(a\right)$ satisfies it i.e. $a+b+c+1=7$