How do you apply a Butcher tableau for an implicit method with matrices? I'm a little bit confused about how to understand Butcher tableaus for implicit methods, when I have a matrix. Say I have some ODE that is defined by
$$
\frac{\partial \mathbf{y}}{\partial t} = \underline{\mathbf{M}}\mathbf{y}
$$
where $\mathbf{y}$ is a vector and $\underline{\mathbf{M}}$ is a constant matrix.
I can write out the implicit Euler method like
$$
\frac{\mathbf{y}^{n+1}-\mathbf{y}^n}{h}=\underline{\mathbf{M}}\mathbf{y}^{n+1}
$$
which gives as solution for $\mathbf{y}^{n+1}$
$$
\mathbf{y}^{n+1}=(\underline{\mathbf{1}}-\underline{\mathbf{M}})^{-1}\cdot\mathbf{y}^n
$$
However if I follow the Butcher tableau for backward Euler (here: https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods#Backward_Euler), I end up with the following scheme
$$
y^{n+1}=y^n + hk_1 \\
k_1 = f(t_n + h, y_n+hk_1)
$$
If I apply this to my problem, then $\mathbf{k_1}=\underline{\mathbf{M}}\cdot(\mathbf{y}^n+h\mathbf{k_1})$, but then rearranging this for $k_1$ gives $\mathbf{k_1}=(\underline{\mathbf{1}}-\underline{\mathbf{M}})^{-1}\cdot\underline{\mathbf{M}}\cdot\mathbf{y}^n$, so
$$
\mathbf{y}^{n+1}=\mathbf{y}^n + h(\underline{\mathbf{1}}-\underline{\mathbf{M}})^{-1}\cdot\underline{\mathbf{M}}\cdot\mathbf{y}^n
$$
which is a different update for $\mathbf{y}^{n+1}$ that I got before. What misconception have I made here? Am I applying the Butcher tableau wrong?

You have some errors in your transformations.
$\newcommand{\vv}[1]{\mathbf{#1}}\newcommand{vk}{\vv{k}}\newcommand{vy}{\vv{y}}$
$\newcommand{\mm}[1]{\underline{\mathbf #1}}\newcommand{\mM}{\mm{M}}\newcommand{\mI}{\mm{I}}$
The direct Euler formula gives $$\vy^{n+1}=(\mI-h\mM)^{-1}·\vy^n.$$
With the stage formulation you likewise get
$$
\vk_1=\mM·(\mI-h\mM)^{-1}·\vy^n
$$
and thus
$$
\vy^{n+1}=\vy^n+h\vk_1=(\mI-h\mM)^{-1}·[(\mI-h\mM)·\vy^n+h\mM·\vy^n]=(\mI-h\mM)^{-1}·\vy^n.
$$
so indeed the same result.
