matrices too big to fit the width of a page

I suggest you insert, after the \begin{frame} statement but before the start of the equation, the following instructions.

\footnotesize
\setlength{\arraycolsep}{2.5pt}
\medmuskip = 1mu % default: 4mu plus 2mu minus 4mu

The command \footnotesize will bring about a roughly 20% reduction in fontsize; \arraycolsep=3pt cuts about 40% from the whitespace that gets inserted between columns (useful because you have 7 such whitespaces), and \medmuskip=1mu drastically reduces the amount of whitespace TeX inserts before and after each "+" symbol (and all other symbols of class "mathbin").

Here's a complete MWE and its output:

\documentclass{beamer}
\begin{document}
\begin{frame}
\footnotesize
\setlength{\arraycolsep}{2.5pt} % default: 5pt
\medmuskip = 1mu % default: 4mu plus 2mu minus 4mu
\[ 
M_{1,k}  = \frac{1}{15} 
\left( \begin{array}{rrrrrr} 
0 & 0 & 0 & 0 & 0 & 0  \\
0 & 12k b_0 & (12k+15) b_1 & (12k+30) b_2 & (12k+45) b_3 & (12k+60) b_4 \\
0 & 0 & 0 & 0 & 0 & 0   \\
0 & 2k  a_0 & (2k+5)  a_1 & (2k+10)  a_2 & 0 & 0   \\
\end{array} \right)
\]
\end{frame}
\end{document}

Addendum: If you need to economize further on whitespace, you could (i) reduce the whitespace around the equal sign by issuing the command \thickmuskip = 2mu and (ii) replace the array preamble line \begin{array}{rrrrrr} with \begin{array}{@{}rrrrrr@{}}; the latter measure will eliminate the whitespace between the opening and closing large parentheses and the first/last column of numbers.

One quick (but not very beautiful) solution is to work with \scalebox from the graphicx package, e.g.

\newcommand\scalemath[2]{\scalebox{#1}{\mbox{\ensuremath{\displaystyle #2}}}}

\[
M_{1,k}  = \frac{1}{15} \left(
    \scalemath{0.3}{
    \begin{array}{cccccccc}
     0 & 0 & 0 & 0 & 0 & 0  \\
     0 & 12k b_0 & (12k+15) b_1 & (12k+30) b_2 & (12k+45)  b_3 & (12k+60) b_4  \\
     0 & 0 & 0 & 0 & 0 & 0   \\
     0 & 2k  a_0 & (2k+5)  a_1 & (2k+10)  a_2 & 0 & 0   \\
    \end{array}
    }
  \right)
\]

Resize the width to \linewidth

\documentclass{beamer}
\begin{document}
\begin{frame}

\resizebox{\linewidth}{!}{%
$\displaystyle
M_{1,k}  = \frac{1}{15} 
\left( \begin{array}{rrrrrrrr} 
0 & 0 & 0 & 0 & 0 & 0  \\
0 & 12k b_0 & (12k+15) b_1 & (12k+30) b_2 & (12k+45)  b_3 & (12k+60) b_4  \\
0 & 0 & 0 & 0 & 0 & 0   \\
0 & 2k  a_0 & (2k+5)  a_1 & (2k+10)  a_2 & 0 & 0   \\
\end{array} \right)
$}
\end{frame}

\end{document}