Matrix.AddSpin($M$) takes a Matrix $M$ that acts on a space with no spin included and returns the matrix doubled in size, where every former entry now splits in spin-up and spin-down. We assume the ordering of the basis alternates between spin up and down. This behaviour is equivalent to calling Matrix.Enlarge($M$,2).
A = Matrix.New({{1,2},{3,4}}) B = Matrix.AddSpin(A) print(B)
{ { 1 , 0 , 2 , 0 } ,
{ 0 , 1 , 0 , 2 } ,
{ 3 , 0 , 4 , 0 } ,
{ 0 , 3 , 0 , 4 } }