Question

Solve for $x$,$\left|\begin{array}{ccc}x& -6& -1\\ 2& -3x& x-3\\ -3& 2x& x+2\end{array}\right|=0$. Find sum of all values of x.

Answer 1

$\left|\begin{array}{ccc}x& -6& -1\\ 2& -3x& x-3\\ -3& 2x& x+2\end{array}\right|=0$
applying $R_1\to R_1-R_2$ gives
$\to (x-2)\left|\begin{array}{ccc}1& 3& -1\\ 2& -3x& x-3\\ -3& 2x& x+2\end{array}\right|=0$
applying $C_3\to C_1+C_3$ gives
$\to (x-2)(x-1)\left|\begin{array}{ccc}1& 3& 0\\ 2& -3x& 1\\ -3& 2x& 1\end{array}\right|=0$
$\Rightarrow (x-2)(x-1)(-5x-15)=0$
$\Rightarrow (x-1)(x-2)(x+3)=0$
$\therefore$ Possible values of $x$ are $-3,1,2$
Hence, sum of all values of $x=-3+1+2=0$