Question

Evaluate $\left|\begin{array}{ccc}x& 1& 1\\ 1& x& 1\\ 1& 1& x\end{array}\right|$

Answer 1

$$\begin{gathered} \mathrm{Let}∆=\left|\begin{array}{ccc}x& 1& 1\\ 1& x& 1\\ 1& 1& x\end{array}\right| \\ =\left|\begin{array}{ccc}x-1& 1-x& 0\\ 1& x& 1\\ 1& 1& x\end{array}\right|\left[\mathrm{Applying}{R}_1\to R_1-R_2\right] \\ =\left(x-1\right)\left|\begin{array}{ccc}1& -1& 0\\ 1& x& 1\\ 1& 1& x\end{array}\right| \\ =\left(x-1\right)\left|\begin{array}{ccc}1& 0& 0\\ 1& x+2& 1\\ 1& x+2& x\end{array}\right|\left[\mathrm{Applying}{C}_2\to C_1+C_2+C_3\right] \\ =\left(x-1\right)\left(x+2\right)\left|\begin{array}{ccc}1& 0& 0\\ 1& 1& 1\\ 1& 1& x\end{array}\right|\left[\mathrm{Expanding}\mathrm{along}{R}_1\right] \\ =\left(x-1\right)\left(x+2\right)\left(x-1\right) \\ ∆={\left(x-1\right)}^2\left(x+2\right) \end{gathered}$$