# Multiline equations with matrix and aligned regular equation underneath

I would like to write an equation consisting of one equation with matrices and one equation referencing the above matrices underneath (see picture).
What I tried can be seen in a minimized version in the code. It throws error 'Extra alignment tab has been changed to \cr'. When i skip the error multiple times every column stands in a own row.

$$\label{eq:NewEul}\begin{split} \begin{bmatrix} M1 & & 0 \\ & \ddots & \\ 0 & & Mn \end{bmatrix} & \begin{bmatrix} a1 \\ \vdots \\ an \end{bmatrix} & = & \begin{bmatrix} f1 \\ \vdots \\ fn \end{bmatrix} \\ M & a & = & f \end{split}$$


My question is: How do I need to modify my code regarding the alignment to match the style shown in the picture?

\documentclass{article}
\usepackage{bm}
\usepackage{amsmath}
\usepackage{booktabs}

% for show just the equation ...
\usepackage[active,displaymath,tightpage]{preview}
\setlength\PreviewBorder{1em}

\begin{document}
$$\arraycolsep=0pt \label{eq:NewEul} \begin{array}{*{8}{c}} \begin{bmatrix} \widehat{M}_1 & & 0 \\ & \ddots & \\ 0 & & \widehat{M}_n \end{bmatrix} & \begin{bmatrix} \hat{a}_1 \\ \vdots \\ \hat{a}_n \end{bmatrix} & = & \begin{bmatrix} \hat{f}_1^c \\ \vdots \\ \hat{f}_n^c \end{bmatrix} & + & \begin{bmatrix} \hat{f}_1^e \\ \vdots \\ \hat{f}_n^e \end{bmatrix} & + & \begin{bmatrix} \hat{f}_1^r \\ \vdots \\ \hat{f}_n^r \end{bmatrix} \\ \addlinespace \bm{\widehat{M}} & \bm{\hat{a}} & = & \bm{\hat{f}_1^c} & + & \bm{\hat{f}_1^e} & + & \bm{\hat{f}_1^r} \end{array}$$
\end{document}