Question

The value of $\left|\begin{array}{ccc}a+x& y& z\\ x& a+y& z\\ x& y& a+z\end{array}\right|$ is equal to

• A. $0$
• B. $a^2(a+z+x+y)$
• C. $a^2(z+x+y)$
• D. $a\cdot (a+z+x+y)$

Answer 1

The correct option is B $a^2(a+z+x+y)$

We have, $\left|\begin{array}{ccc}a+x& y& z\\ x& a+y& z\\ x& y& a+z\end{array}\right|$
Applying $C_1\to C_1+C_2+C_3=\left|\begin{array}{ccc}a+x+y+z& y& z\\ a+x+y+z& a+y& z\\ a+x+y+z& y& a+z\end{array}\right|=(a+x+y+z)\left|\begin{array}{ccc}1& y& z\\ 1& a+y& z\\ 1& y& a+z\end{array}\right|$
Applying $R_2\to R_2-R_1\text{and}{R}_3\to R_3-R_1=(a+x+y+z)\left|\begin{array}{ccc}1& y& z\\ 0& a& 0\\ 0& 0& a\end{array}\right|=(a+x+y+z)\left|\begin{array}{cc}a& 0\\ 0& a\end{array}\right|=a^2(a+z+x+y)$