Question

lf $\left|\begin{array}{ccc}a+x& a-x& a-x\\ a-x& a+x& a-x\\ a-x& a-x& a+x\end{array}\right|=0$ then the non-zero value of x=............

• A. $a$
• B. $3a$
• C. $2a$
• D. $4a$

Answer 1

The correct option is B $3a$

By operation of matrix (5),
$A=\left|\begin{array}{ccc}a+x& a-x& a-x\\ a-x& a+x& a-x\\ a-x& a-x& a+x\end{array}\right|$
$=(a+x)\left[\right(a+x{)}^2-(a-x{)}^2]-(a-x)[a^2-x^2-(a-x{)}^2]+(a-x\left)\right[(a-x{)}^2-(a^2-x^2\left)\right]$
$=(a+x)\left(4ax\right)-(a-x)(a^2-x^2)+2(a-x{)}^3-(a-x\left)\right(a^2-x^2)$
$=(a+x)\left(4ax\right)+2(a-x)[a^2+x^2-2ax-a^2+x^2]$
$=(a+x)4ax+2(a-x)(2x^2-2ax)$
$A=4a^{2}x+4a{x}^2+4a{x}^2-4a^{2}x-a{x}^3+4a{x}^2$
$A=8a{x}^2+4a{x}^2-4x^3$
$A=12a{x}^2-4x^3$
Given,$A=12a{x}^2-4x^3=0$
$x^2(12a-x)=0$
$x=3a$