I have the following set of equations: \begin{align} u_{s}=-\beta \left ( u_{on} - u_{br}\right ) + \gamma u_{l} + v_{s}, \end{align} \begin{align} u_{l}=\lambda \left ( u_{on} - u_{br} \right ) + v_{l}, \end{align} \begin{align} u_{a}=\phi_{l}v_{l} + \phi_{s}v_{s} + v_{a}. \end{align} Assume $u_{br} = 0$ and $u_{a} = u_{l} + u_{s}$. Rewrite the model in the form \begin{align} \mathbf{\left ( I - G_{0} \right ) u^{p} = A^{p}v^{p}}. \end{align} Substitute out $u_{s}$ in equation (1) to get $$\mathbf{\left ( I - G_{0} \right ) u^{p}} = \begin{bmatrix} (1-\gamma) &1 &\beta \\ 0&1 &0 \\ 1&0 &-\lambda \end{bmatrix} \begin{bmatrix} u_{l}\\ u_{a}\\ u_{on} \end{bmatrix}.$$
Also, $$\mathbf{A^{p}v^{p}} = \begin{bmatrix} 0 &0 &1 \\ \phi_{s} &1 &\phi_{l} \\ 1&0 &0 \end{bmatrix} \begin{bmatrix} v_{l}\\ v_{a}\\ v_{s} \end{bmatrix}.$$
EDIT: The determinant of $\mathbf{\left ( I - G_{0} \right )} = \lambda \left ( 1 - \gamma \right ) - \beta,$ and using the method of determinants and cofactors, $$\mathbf{\left ( I - G_{0} \right )^{-1}} = \frac{1}{\lambda\left ( 1 - \gamma \right ) - \beta} \begin{bmatrix} -\lambda &\lambda &-\beta \\ 0&-\lambda\left ( 1-\gamma \right )-\beta &0 \\ -1&1 &1-\gamma \end{bmatrix}.$$
After some tedious matrix algebra, I find that $$\mathbf{\left ( I - G_{0} \right )^{-1} A^{p}v^{p}} = \frac{1}{\lambda\left ( 1 - \gamma \right ) - \beta} \begin{bmatrix} \phi_{s} \lambda - \beta&\lambda & \lambda\left ( \phi_{l} - 1 \right )\\ \phi_{s}\left ( -\lambda\left ( 1-\gamma \right )- \beta \right )&-\lambda\left ( 1-\gamma \right )-\beta & \phi_{l}\left ( -\lambda\left ( 1-\gamma \right )- \beta \right )\\ \phi_{s}-1-\gamma&1 &\phi_{l}-1 \end{bmatrix},$$ and \begin{align*} u^{p} = \begin{bmatrix} u_{l} &u_{a} &u_{on} \end{bmatrix}^{'}, \end{align*} \begin{align*} v^{p} = \begin{bmatrix} v_{l} &v_{a} &v_{s} \end{bmatrix}^{'}. \end{align*} What I am having trouble with is how the expression below was obtained or what result is correct.
\begin{align*} \mathbf{\left ( I - G_{0} \right )^{-1} A^{p}} = \frac{1}{\lambda (\gamma -1)}\begin{bmatrix} \beta -\lambda \phi_{l} &-\gamma & \lambda \left ( 1 - \phi _{s} \right )\\ \phi_{l}\left ( \beta + \lambda \left ( \gamma -1 \right ) \right )&\beta + \lambda \left ( \gamma -1 \right ) &\phi_{s}\left ( \beta + \lambda \left ( \gamma -1 \right ) \right ) \\ 1 - \gamma - \phi_{l} &-1 &1 - \phi_{s} \end{bmatrix}, \end{align*} and \begin{align*} u^{p} = \begin{bmatrix} u_{l} &u_{a} &u_{o} \end{bmatrix}^{'}, \end{align*} \begin{align*} v^{p} = \begin{bmatrix} v_{l} &v_{a} &v_{s} \end{bmatrix}^{'}. \end{align*}
Do I have the correct result of deriving $\mathbf{\left ( I - G_{0} \right )^{-1} A^{p}}$?