Pensieve header: A mini implementation of the Gassner matrices.

U[n_,i_,t_] := ReplacePart[
IdentityMatrix[n],
{{i,i}->1-t, {i,i+1}->t,
{i+1,i}->1, {i+1,i+1}->0}
];

MatrixForm /@ (ms1={U[3,1,(Subscript[t, 2])], U[3,2,(Subscript[t, 3])], U[3,1,(Subscript[t, 3])]})

Dot@@ms1 // Simplify // MatrixForm

MatrixForm /@ (ms2={U[3,2,(Subscript[t, 3])], U[3,1,(Subscript[t, 3])], U[3,2,(Subscript[t, 2])]})

Dot@@ms2 // Simplify // MatrixForm