[Tugindia] Suggestions requested

[mkalidoss]

I have appended a tex file below; this file is used 
to produce three lines of output.
1. In the first line I have used \cdot 3; is this correct?
    Or should we use .3?
2. In the second line, to give the limits to \lim, I have
    used \limits after declaring \mathop{\lim}.  Is this 
    correct?
3. In the third line, LHS of the equation seems to be 
    OK, while on the RHS, the alignment of y_1 and y_2
    is not suitable. 
I shall be thankful to receive comments from others.
Sincerely,
M. Kalidoss 

---------- beginning of the sample file------------------------------------------- 
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[
M_R=\mathop{%
\begin{bmatrix}1&1\\0&0\end{bmatrix}
}%
\limits_{R_{.4}}
=\begin{matrix}x\\y\end{matrix}%
\mathop{%
\begin{bmatrix}\cdot 3&\cdot 2\\0&1\end{bmatrix}
}%
\limits^{%
\begin{matrix}a&b\end{matrix}}
\]
\[
\mathop{\lim}\limits_{w \rightarrow \infty}(a^w+b^w)^{1/w}=\max(a,b).
\]
\[
\begin{matrix}x_1\\x_2\end{matrix}
\mathop{%
\begin{bmatrix}1&1\\1&1\end{bmatrix}
}% 
\limits^{%
\begin{matrix}y&y\end{matrix}}
=
\begin{matrix}x_1\\x_2\end{matrix}
\mathop{%
\begin{bmatrix}1&1\\1&1\end{bmatrix}
}% 
\limits^{%
\begin{matrix}y_1&y_2\end{matrix}}
\]
\end{document}
---------- end of the sample file---------------------------------------------------