Step 1 :\(\textbf{Part (a)}\)
Step 2 :\(\begin{vmatrix} 1 & -1 & 0 \ 2 & 0 & 1 \ 3 & 1 & 2 \ \end{vmatrix}\)
Step 3 :\(= 1 \begin{vmatrix} 0 & 1 \ 1 & 2 \ \end{vmatrix} - (-1) \begin{vmatrix} 2 & 1 \ 3 & 2 \ \end{vmatrix} + 0 \begin{vmatrix} 2 & 0 \ 3 & 1 \ \end{vmatrix}\)
Step 4 :\(= 1(0 - 2) - (-1)(4 - 3)\)
Step 5 :\(= -2 + 1\)
Step 6 :\(\boxed{-1}\)
Step 7 :\(\textbf{Part (b)}\)
Step 8 :\(\begin{vmatrix} -1 & 3 & 1 \ 1 & 3 & 2 \ 2 & 1 & 2 \ \end{vmatrix}\)
Step 9 :\(= -1 \begin{vmatrix} 3 & 2 \ 1 & 2 \ \end{vmatrix} - 3 \begin{vmatrix} 1 & 2 \ 2 & 2 \ \end{vmatrix} + 1 \begin{vmatrix} 1 & 3 \ 2 & 1 \ \end{vmatrix}\)
Step 10 :\(= -1(6 - 2) - 3(2 - 4) + 1(1 - 6)\)
Step 11 :\(= 4 + 6 - 5\)
Step 12 :\(\boxed{5}\)
Step 13 :\(\textbf{Part (c)}\)
Step 14 :\(\begin{vmatrix} 1 & 0 \ -1 & 1 \ \end{vmatrix}\)
Step 15 :\(= 1(1) - 0(-1)\)
Step 16 :\(\boxed{1}\)
Step 17 :\(\textbf{Part (d)}\)
Step 18 :\(\begin{vmatrix} 1 & 1 \ 1 & 1 \ \end{vmatrix}\)
Step 19 :\(= 1(1) - 1(1)\)
Step 20 :\(\boxed{0}\)