Z to F

Inverse[(({{1, -z11}, {0, z21}}))].(({{0, -z12}, {1, z22}})) // Simplify

F to Z

Inverse[(({{1, -a}, {0, c}}))].(({{0, -b}, {1, d}})) // Simplify

S to T

t=Inverse[(({{1, -s11}, {0, s21}}))].(({{s12, 0}, {-s22, 1}})) // Simplify

T to S

s=Inverse[(({{1, -t12}, {0, t22}}))].(({{0, t11}, {1, -t21}})) // Simplify

Zin of TL

Solve[z0*((1+x)/(1-x))==zl,x]

Cosh[x] //TrigToExp
Sinh[x] //TrigToExp
Tanh[x] //TrigToExp //Simplify

Cosh[I*β ]//ComplexExpand // Simplify 
Sinh[I*β ]//ComplexExpand // Simplify 
Tanh[I*β ]//ComplexExpand // Simplify

Z matrix of TL

((Sqrt[z0])*(({{Exp[γ*l], Exp[-γ*l]}, {1, 1}}))).Inverse[((1/(Sqrt[z0]))*(({{Exp[γ*l], -Exp[-γ*l]}, {-1, 1}})))] // ExpToTrig//Simplify // MatrixForm

(({{z0 Coth[l γ], z0 Csch[l γ]}, {z0 Csch[l γ], z0 Coth[l γ]}})) /. γ->I*β //ComplexExpand // Simplify // MatrixForm