Question

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

• A. $-\frac{1}{3}(\sum a)$
• B. $-\frac{1}{3}(\sum ab)$
• C. $-abc$
• D. $\sum ab$

Answer 1

Answer: (A)

$-\frac{1}{3}(\sum a)$

$C_1\to C_1+C_2+C_3(a+b+c-Bx)\left|\begin{array}{ccc}1& b+x& c+x\\ 1& c+x& a+x\\ 1& a+x& b+x\end{array}\right|$
$C_1\to R_1-R_2{R}_2\to R_2-R_3(a+b+c-Bx)\left|\begin{array}{ccc}1& b-c& c-a\\ 1& c-a& a-b\\ 1& a+x& b+x\end{array}\right|$
$(a+b+c-Bx)\left[\right(b-c\left)\right(a-b)-(c-a{)}^2]=0$
$\therefore x=-\frac{1}{3}(a+b+c)$
$x=-\frac{1}{3}\sum a$