Question

Using the properties of determinants, solve the following for $x$:

$\left|\begin{array}{ccc}x+2& x+6& x-1\\ x+6& x-1& x+2\\ x-1& x+2& x+6\end{array}\right|=0$

Answer 1

$\left|\begin{array}{ccc}x+2& x+6& x-1\\ x+6& x-1& x+2\\ x-1& x+2& x+6\end{array}\right|=0$

$R_1\to R_1+R_2+R_3$

$\left|\begin{array}{ccc}3x+7& 3x+7& 3x+7\\ x+6& x-1& x+2\\ x-1& x+2& x+6\end{array}\right|=0$

$(3x+7)\left|\begin{array}{ccc}1& 1& 1\\ x+6& x-1& x+2\\ x-1& x+2& x+6\end{array}\right|=0$

$C_2\to C_2-C_1\&C_3\to C_3-C_1$

$(3x+7)\left|\begin{array}{ccc}1& 0& 0\\ x+6& -7& -4\\ x-1& 3& 7\end{array}\right|=0$

Expand along $R_1$

$\Rightarrow (3x+7)(-49+12)=0$

$\Rightarrow (3x+7)(-37)=0$

$\Rightarrow x=\frac{-7}{3}$