Root of the equation x-1 1 1 ,1 x-1 1, 1 1 x-1

Question

Root of the equation \(\begin{vmatrix} x-1&1&1\\1&x-1&1\\1&1&x-1\end{vmatrix} = 0\)

Answer 1

\(\begin{vmatrix} x-1&1&1\\1&x-1&1\\1&1&x-1\end{vmatrix} = 0\)

\(R_2 \to R_2 - R_3\)

\(R_3 \to R_3 - R_1\)

\(\begin{vmatrix} x-1&1&1\\0&x-2&2-x\\2-x&0&x-2\end{vmatrix} = 0\)

\((x - 1)((x - 2)^2) -1(-(2- x)^2)+ 1(-(x - 2)(2 -x)) = 0\)

\(x - 1(x^2 + 4 - 4x) + 4 + x^2 - 4x + x^2 + 4 - 4x= 0\)

\(x^3 + 4x - 4x^2 - x^2 -4 + 4x + 8 + 2x^2 - 8x = 0\)

\(x^3- 3x^2 + 4= 0\)

The roots of the following equations are (-1, 2).