How solve this matricial system

Question

I'm trying to solve this matricial system below and had no success. I'm also not understanding why Vin(s) vanish from the system.

Matricial System The system above leads to

Transfer Function

I'm very grateful if anyone help me to solve this system setp-by-step

Answer 1

Inverting the $2\times 2$ matrix gives $$ \begin{bmatrix}sL & 1-D \\ 1-D & -\left(sC+\frac{1}{R}\right)\end{bmatrix}^{-1} =\frac{1}{LCs^2+\frac{L}{R}s+(1-D)^2} \begin{bmatrix}sC+\frac{1}{R} & 1-D \\ 1-D & -sL\end{bmatrix} $$ so $$ \hat{i}_L(s) = \frac{\left[\left(sC+\frac{1}{R}\right)V_0+(1-D)I_L\right]\hat{d}(s)+\left(sC+\frac{1}{R}\right)\hat{v}_{in}(s)}{LCs^2+\frac{L}{R}s+(1-D)^2}. $$ Now applying a hidden constraint that $(sC+\frac{1}{R})\hat{v}_{in}(s)=\left[(1-D)I_L-\frac{1}{R}V_0\right]\hat{d}(s)$ gives the desired result.