VA={0,1/Sqrt[3]};

VB={1/2,-1/(2 Sqrt[3])};

VC={-1/2,-1/(2 Sqrt[3])};

len[x_]=x.x

len[VA-VB]

len[VA-VC]

len[VB-VC]

Iden={{1,0},{0,1}};

A=1/2{{-1,-Sqrt[3]},{Sqrt[3],-1}}

Simplify[A.VB]

B=1/2{{-1,Sqrt[3]},{-Sqrt[3],-1}};

F={{-1,0},{0,1}};

G=1/2{{1,-Sqrt[3]},{-Sqrt[3],-1}};

H=1/2{{1,Sqrt[3]},{Sqrt[3],-1}};

MatrixForm[VA]

MatrixForm[VB]

MatrixForm[VC]

MatrixForm[A]

MatrixForm[B]

MatrixForm[F]

MatrixForm[G]

MatrixForm[H]

MatrixForm[Simplify[A.VA]]

MatrixForm[Simplify[B.VA]]

MatrixForm[Simplify[F.VA]]

MatrixForm[Simplify[G.VA]]

MatrixForm[Simplify[H.VA]]

MatrixForm[Simplify[Iden.VA]]