Question

Solve the following equation by using properties of determinants $\left|\begin{array}{c}x+2\\ x+6\\ x-1\end{array}\begin{array}{c}x+6\\ x-1\\ x+2\end{array}\begin{array}{c}x-1\\ x+2\\ x+6\end{array}\right|=0$

Answer 1

$\begin{array}{c}\left|\begin{array}{ccc}x+2& x+6& x-1\\ x+6& x-1& x+2\\ x-1& x+2& x+6\end{array}\right|=0\\ R_1\to R_1+R_2+R_3\\ \left|\begin{array}{ccc}3x+7& 3x+7& 3x+7\\ x+6& x-1& x+2\\ x-1& x+2& x+6\end{array}\right|=0\\ \\ \left(3x+7\right)\left|\begin{array}{ccc}1& 1& 1\\ x+6& x-1& x+2\\ x-1& x+2& x+6\end{array}\right|=0\\ C_2\to C_2-C_1{C}_3\to C_3-C_1\\ \left(3x+7\right)\left|\begin{array}{ccc}1& 0& 0\\ x+6& -7& -4\\ x-1& 3& 7\end{array}\right|=0\\ \left(3x+7\right)\left(-49+12\right)=0\\ x=\frac{-7}{3}\end{array}$

Hence, this is the answer.