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\tilde{A}(0)\\  0 \\   \end{bmatrix}\]  Carrying the matrix multiplication step by step \[\begin{bmatrix} \tilde{A}(z)\\  \tilde{B}(z)\\  \end{bmatrix}=\begin{bmatrix}  \frac{\beta_c cos \beta_c(z)-i\delta sin \beta_c(z)}{\beta_c}e^{i \delta (z)} & \frac{i \kappa_{ab}}{\beta_c}sin\beta_c(z)e^{i \delta (z)}\\  \frac{i \kappa_{ba}}{\beta_c}sin\beta_c(z)e^{-i \delta (z)} &\frac{\beta_c cos \beta_c(z)+i\delta sin \beta_c(z)}{\beta_c}e^{-i \delta (z)} \\   \end{bmatrix} \]