\(\begin{vmatrix}-1 & -1 & -1\\3 & 1 & -1\\1 & 2 & 1\end{vmatrix}\) = -\(\begin{vmatrix}1 & -1 \\2 & 1\end{vmatrix}\) + \(\begin{vmatrix}3 & -1\\1 & 1\end{vmatrix}\) - \(\begin{vmatrix}3 & 1\\1 & 2\end{vmatrix}\) = -( 1 + 2) + (3 + 1) - ( 6 - 1) = -3 + 4 - 5 = 4 - 8 = -4.