Question

The non zero absolute value of $x$ for which
$\left|\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right|=0$ is:

Answer 1

Given
$\left|\begin{array}{ccc}3& x-1& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right|=0$

$\Rightarrow \left|\begin{array}{ccc}0& x& -x\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right|=0[R_1\to R_1-R_2]$

$\Rightarrow \left|\begin{array}{ccc}0& x& -x\\ -x& 0& x\\ x+3& -1& 2\end{array}\right|=0[R_2\to R_2-R_3]$

$\Rightarrow \left|\begin{array}{ccc}0& x& 0\\ -x& 0& x\\ x+3& -1& 1\end{array}\right|=0[C_3\to C_3+C_2]$

$\Rightarrow -x\left[\right(-x)-x(x+3\left)\right]=0\Rightarrow x(x^2+4x)=0$
$\Rightarrow x=0,-4$
Hence non zero value of $x$ is $x=-4$