Question

Evaluate the following:

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

Answer 1

Let $∆=\left|\begin{array}{ccc}x& 1& 1\\ 1& x& 1\\ 1& 1& x\end{array}\right|$.
$$\begin{gathered} ∆=\left|\begin{array}{ccc}x& 1& 1\\ 1& x& 1\\ 1& 1& x\end{array}\right| \\ =\left|\begin{array}{ccc}x-1& 1-x& 0\\ 1& x& 1\\ 0& 1-x& x-1\end{array}\right|\left[\mathrm{Applying}{R}_1\to R_1-R_2\mathrm{and}{R}_3\to R_3-R_2\right] \\ ={\left(x-1\right)}^2\left|\begin{array}{ccc}1& -1& 0\\ 1& x& 1\\ 0& -1& 1\end{array}\right| \\ ={\left(x-1\right)}^2\left|\begin{array}{ccc}1& -1& 0\\ 1& x+1& 1\\ 0& 0& 1\end{array}\right|\left[\mathrm{Applying}{C}_2\to C_2+C_3\right] \\ ={\left(x-1\right)}^2(x+1+1)\left[\mathrm{Expanding}\mathrm{along}\mathrm{last}\mathrm{row}\right] \\ ={\left(x-1\right)}^2(x+2) \\ \therefore ∆={\left(x-1\right)}^2(x+2) \end{gathered}$$