I would like to apply a strategy I have used in solving the following bilinear system of equations to solving a related system in matrix form. Consider this system of equations: \begin{equation} \begin{split} \Lambda-\frac{\beta x y}{x+y+z}-\mu x + \gamma z=0\\ \frac{\beta x y}{x+y+z}-(\alpha + \kappa+\mu)y=0\\ \kappa y - (\mu+\gamma)z=0 \end{split} \end{equation} to solve for $x$, $y$ and $z$, we begin from the last equation where we notice that $$z=\frac{\kappa}{\mu + \gamma}y$$
Now, from the second equation, we have $$ x+y+z=\frac{\beta x }{\alpha+\kappa+\mu}\Rightarrow y=\frac{\beta x }{\alpha+\kappa+\mu}-x-z\Rightarrow y=\frac{\beta x }{\alpha+\kappa+\mu}-x-\frac{\kappa}{\mu + \gamma}y $$ Where the last term is after substitution of the value of $z$ obtained from equation 3. We therefore have that $$ y=\frac{(\mu + \gamma)(r-1)}{(\mu+\gamma+\kappa)}x\hspace{3mm}\text{and}\hspace{3mm}x=\frac{(\mu+\gamma+\kappa)}{(\mu + \gamma)(r-1)}y $$ where $r=\frac{\beta}{\alpha+\kappa+\mu}$. Substituting the values of $x$, $y$ and $z$ and $x+y+z$ in the first equation, we obtain $$ \Lambda-\bigg[(\alpha+\kappa+\mu)+\frac{\mu(\mu+\gamma+\kappa)}{(\mu+\gamma)(r-1)}-\frac{\gamma\kappa}{\mu+\gamma}\bigg]y=0 $$ which after some algebra, reduces to $$ \Lambda-\bigg[\frac{\alpha(\mu+\gamma)(r-1)+\mu(\mu+\gamma+\kappa)r}{(\mu+\gamma)(r-1)}\bigg]y=0 $$ We therefore are able to obtain the explicit values of $x$, $y$ and $z$ as
\begin{equation} \begin{split} x=\frac{\Lambda(\mu+\gamma+\kappa)}{\alpha(\mu+\gamma)(r-1)+\mu(\mu+\gamma+\kappa)r}\\ y=\frac{\Lambda(\mu+\gamma)(r-1)}{\alpha(\mu+\gamma)(r-1)+\mu(\mu+\gamma+\kappa)r}\\ z=\frac{\Lambda\kappa(r-1)}{\alpha(\mu+\gamma)(r-1)+\mu(\mu+\gamma+\kappa)r} \end{split} \end{equation}
I would like to apply the above strategy to solve the following $n$ system of equations for $x$, $y$ and $z$ which is in matrix form. It appears simple but unfortunately, I am not able to proceed, especially starting from the last equation to the first equation as I have done above.
Let $\boldsymbol{\beta}=[(\beta_{ij})]_{n\times n }$ be a matrix of constants and $x=[x_{1},\cdots,x_{n}]^{T}$, $y=[y_{1},\cdots,y_{n}]^{T}$ and $z=[z_{1},\cdots,z_{n}]^{T}$ be vectors of variables. Also, let $\Lambda=[\Lambda_{1},\cdots,\Lambda_{n}]^{T}$, $\gamma=[\gamma_{1},\cdots,\gamma_{n}]^{T}$, $\mu=[\mu_{1},\cdots,\mu_{n}]^{T}$, $\tau=[\tau_{1},\cdots,\tau_{n}]^{T}$ and $\psi=[\psi_{1},\cdots,\psi_{n}]^{T}$ be vectors of constants. Solve for $x$, $y$ and $z$ in the following system
\begin{equation} \begin{split} \Lambda-\text{diag}(x)\boldsymbol{\beta}\text{diag}^{-1}(x+y+z)y-\text{diag}(\mu)x+\text{diag}(\tau)z=0\\ \text{diag}(x)\boldsymbol{\beta}\text{diag}^{-1}(x+y+z)y-\text{diag}(\gamma+\mu+\psi)y=0\\ \text{diag}(\gamma)y-\text{diag}(\tau+\mu)z=0 \end{split} \end{equation} Notice that $0$ is and $n\times 1$ vector of zeros. I will appreciate it so much if somebody would help me solve this system.