Question

The value of $\left|\begin{array}{ccc}1+x& 2& 3\\ 1& 2+x& 3\\ 1& 2& 3+x\end{array}\right|$ is

• A. $(x-6)x^2$
• B. $(x-6)x$
• C. $x^2(x+6)$
• D. $(x-6)$

Answer 1

Answer: (C)

$x^2(x+6)$

$\left|\begin{array}{ccc}1+x& 2& 3\\ 1& 2+x& 3\\ 1& 2& 3+x\end{array}\right|$
$R_1\to R_1-R_2$ and $R_2\to R_2-R_3$
$\left|\begin{array}{ccc}x& -x& 0\\ 0& x& -x\\ 1& 2& 3+x\end{array}\right|$
$=x(x^2+3x-(-2x\left)\right)+1\left(x^2\right)$ ....... [Expanding along $C_1$]
$=x(x^2+5x)+x^2$
$=x^3+6x^2$
$=x^2(x+6)$