Linear Algebra/Determinants Exist/Solutions

Solutions
These summarize the notation used in this book for the $$2$$- and $$3$$- permutations. $$ \begin{array}{c|cc} i         &1      &2    \\ \hline \phi_1(i) &1      &2     \\ \phi_2(i) &2      &1 \end{array} \qquad \begin{array}{c|ccc} i         &1     &2   &3    \\ \hline \phi_1(i) &1     &2   &3    \\ \phi_2(i) &1     &3   &2    \\ \phi_3(i) &2     &1   &3    \\ \phi_4(i) &2     &3   &1    \\ \phi_5(i) &3     &1   &2    \\ \phi_6(i) &3     &2   &1 \end{array} $$