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