Pensieve header: The Gauss-Gassner-Alexander scattering.

Simplify[x1 /. Solve[{
x1==a11 y1+a21 y2+a31 y3,
x2==a12 y1+a22 y2+a32 y3,
x3==a13 y1+a23 y2+a33 y3,
x1==y2,x2==0,y3==1},
{x1,x2,x3,y1,y2,y3}]]

Simplify[x2 /. Solve[{
x1==a11 y1+a21 y2+a31 y3,
x2==a12 y1+a22 y2+a32 y3,
x3==a13 y1+a23 y2+a33 y3,
x2==y3,x1==0,y2==1},
{x1,x2,x3,y1,y2,y3}]]

Simplify[x1 /. Solve[{
x1==a11 y1+a21 y2+a31 y3,
x2==a12 y1+a22 y2+a32 y3,
x3==a13 y1+a23 y2+a33 y3,
x1==y2,x2==1,y3==0},
{x1,x2,x3,y1,y2,y3}]]

Simplify[x2 /. Solve[{
x1==a11 y1+a21 y2+a31 y3,
x2==a12 y1+a22 y2+a32 y3,
x3==a13 y1+a23 y2+a33 y3,
x2==y3,x1==1,y2==0},
{x1,x2,x3,y1,y2,y3}]]

(({{a11, a12, a13}, {a21, a22, a23}, {a31, a32, a33}})) = Inverse[(({{b11, b12, b13}, {b21, b22, b23}, {b31, b32, b33}}))]

{(a11/(a12-a12 a21+a11 a22)),(a12/(a11+a12 a31-a11 a32))}//Simplify

Simplify[x1 /. Solve[{
x1==a11 y1+a21 y2+a31 y3,
x2==a12 y1+a22 y2+a32 y3,
x3==a13 y1+a23 y2+a33 y3,
x1==y2,x2==-1,y3==1},
{x1,x2,x3,y1,y2,y3}]]

Simplify[x2 /. Solve[{
x1==a11 y1+a21 y2+a31 y3,
x2==a12 y1+a22 y2+a32 y3,
x3==a13 y1+a23 y2+a33 y3,
x2==y3,x1==-1y2==1},
{x1,x2,x3,y1,y2,y3}]]