Bad math environment delimiter when using \begin{equation} and \end{equation}

The LaTeX macros \begin{equation} and \[ both initiate a display-math group, and the macros \end{equation} and \] both terminate a display-math group. (In addition, the equation environment provides a method for numbering the equations, whereas \[ ... \] does not.) The LaTeX macros \begin{equation} and \[ are designed purposefully so as not to let users open a display-math group twice; this is why you're getting the error message "Bad math environment delimiter" when LaTeX encounters \[ after having processed \begin{equation}.

The upshot: Use one or the other method for setting up a display-math group, but don't use both simultaneously.

For a more-detailed discussion of how various LaTeX displaymath environments are set up, see this answer to the question "What are the differences between $$, \[, align, equation and displaymath?" Shameless self-citation alert!

I can't improve @Mico 's correct accepted answer to the question you asked, but can suggest that you use a bmatrix environment instead of hard coding the brackets and the arrays. You save typing, improve readability, and minimize fiddling:

\[
\begin{bmatrix}
    v^{0}_{n} \\
    v^{K-1}_{n}
\end{bmatrix} =
    -\frac{1}{C_{ref}} 
\begin{bmatrix}
     V^{0}_{0,n} & \cdots & V^{0}_{M-1,m} \\
        \vdots & \ddots & \vdots \\
        V^{K-1}_{0,n} & \cdots &  V^{K-1}_{M-1,n}
\end{bmatrix}
\begin{bmatrix}
        C_{0,n} \\
        C_{M-1,n}
\end{bmatrix}
\]