How can I visualize Matrix operations?

Can't say about MathJax, but the gauss package does this.

\documentclass{article} 
\usepackage{gauss} 
\begin{document} 
\[
 \begin{gmatrix}[p]
 1 & 2 & 3 \\
 4 & 5 & 6 \\
 7 & 8 & 9
 \rowops
 \swap{0}{1}
 \mult{0}{\cdot 7}
 \add[5]{1}{2}
 \end{gmatrix}
\]
\end{document}

I'm used to the following notation:

You can use the following LaTex Code in MathJax (The code looks a little bit messy because I'm not used to MathJax):

\begin{aligned}
    &\begin{pmatrix}
        1&2&8\\
        1&3&7\\
        2&9&3
    \end{pmatrix}
    \hspace{-0.5em}
    \begin{align}
        &\phantom{I}\\
        &II-I \\
        &III-2\cdot I
    \end{align}
    \newline
    \Rightarrow
    &\begin{pmatrix}
        1&2&8\\
        0&1&-1\\
        0&5&-13
    \end{pmatrix}
\end{aligned}

Since TikZ or PsTricks are not supported in Math JaX getting those back pointing arrows or proper placement of symbols are not possible as far as I know. This is the closest that I can come up within the limitations of Math JaX.

\documentclass[11pt]{amsart}
\usepackage{amsmath}                
\begin{document}
\[\begin{pmatrix}
 1 & 1 & 1\\ 
 t & 2t & 2\\ 
t+1 & 0 & 2t
 \end{pmatrix}\ \hookleftarrow\stackrel{-t}{+}%
 \hookleftarrow\stackrel{-(t+1)}{+}\rightsquigarrow 
\begin{pmatrix}
 1 & 1 & 1\\ 
 0 & t & 2-t\\ 
0 & -t-1 & t-1
 \end{pmatrix}\hookleftarrow+|\cdot(-1)\]
\end{document}