Implicit Runge-Kutta-methods alternative formulation

Question

Implicit Runge-Kutta-methods with $s$ stepts and with constant lenght $h$ for
$$
x'(t)=f(x(t),t), \quad x(t_0)=x_0
$$
have the form
$$
X_i=f(x_{n-1}+h \sum_{j=1}^s a_{ij}X_j,t_{n-1}+hc_i), \quad i=1,…,s \\ x_n=x_{n-1}+h\sum_{i=1}^s b_iX_i
$$
Now my lecture notes state that this system is equivalent to the following one
$$
Z_i=x_{n-1}+h \sum_{j=1}^s a_{ij}f(Z_j,t_{n-1}+hc_i), \quad i=1,…,s \\ x_n=Z_s
$$
with $c_s=1$ and $a_{si}=b_i$ if the Runge-Kutta-Method has order $d\geq1$ and $A=(a_{ij})_{i,j=1}^s$ is regular.

Answer 1

This is just setting $$ Z_i=x_{n-1}+h\sum a_{ij}X_j, $$ where as before $$ X_i=f(Z_i,t_{n-1}+c_ih). $$ If you put the two identities together to eliminate $Z_i$, then you get the first form, if you eliminate $X_i$, then you get the second form.

This is valid for all Runge-Kutta methods, also the explicit ones.