Minors and plucker relations

http://www.math.msu.edu/~magyar/papers/MinorIdentities.pdf

 

\(\Delta _{1,2}\Delta _{12,13}=\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23}\)

1. (mat = Array[Subscript[a, ##] &, {3, 3}]) // MatrixForm
   Minors[mat] // MatrixForm
   Minors[mat, 1] // MatrixForm
   Minors[mat, 2] // MatrixForm
   Minors[mat, 3] // MatrixForm
2. Simplify[Subscript[a, 1,
      3]*(-Subscript[a, 1, 2] Subscript[a, 2, 1] +
        Subscript[a, 1, 1] Subscript[a, 2, 2]) +
     Subscript[a, 1,
      1]*(-Subscript[a, 1, 3] Subscript[a, 2, 2] +
        Subscript[a, 1, 2] Subscript[a, 2, 3])]

 

1. \Delta _{12,12}\text{:=}-a_{1,2} a_{2,1}+a_{1,1} a_{2,2}\Delta _{12,23}\text{:=}-a_{1,3} a_{2,2}+a_{1,2} a_{2,3}\Delta _{1,3}\text{:=}a_{1,1}\Delta _{1,3}\text{:=}a_{1,3}\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23}