Align many matrices the best way possible?

Another possible solution a bit more fast using spalign package:

documentclass[a4paper,12pt]{article}
usepackage[margin=2.2cm]{geometry}
usepackage{mathtools}
usepackage{spalign}

begin{document}
begin{equation}
spalignaugmat[c]{1 a 2 a; 0 1 0 dfrac{-a^2}{a^2-2}; 0 3-a -2 2-a} xrightarrow{begin{matrix}
mathbf{r}_1 rightarrow mathbf{r}_1 - mathbf{r}_2a \
        mathbf{r}_3 rightarrow mathbf{r}_3 - mathbf{r}_2(3-a) 
end{matrix}}{} 
  spalignaugmat{1 0 2 -dfrac{2a}{a^2-2}; 0 1 0 dfrac{a^2}{a^2-2}; 0 0 -2 dfrac{1}{a^2-2}(-a^2+2a-4)}    
end{equation}

begin{equation}
spalignaugmat[c]{1 0 2 -dfrac{2a}{a^2-2}; 0 1 0  dfrac{a^2}{a^2-2}; 0 0 -2 dfrac{1}{a^2-2}(-a^2+2a-4)} xrightarrow{begin{matrix}
mathbf{r}_1 rightarrow mathbf{r}_1 + mathbf{r}_3 \
        mathbf{r}_3 rightarrow -frac{1}{2}mathbf{r}_3
end{matrix}}{}    
  spalignaugmat{1 0 0 -dfrac{a^2+4}{a^2-2}; 0 1 0 dfrac{a^2}{a^2-2}; 0 0 1 dfrac{frac{a^2}{2}-a+2}{a^2-2}}    
end{equation}
[mathbf{S}^{ast}=spalignaugmat[c]{1 0 0 -frac{a^2+4}{a^2-2};0 1 0 frac{a^2}{a^2-2};  0 0 1 frac{frac{a^2}{2}-a+2}{a^2-2}}]

end{document}

documentclass[a4paper]{article}
usepackage[margin=2cm]{geometry}
usepackage{array}
usepackage{amsmath}
begin{document}

begin{align}
    left(begin{array}{ccc|>{displaystyle}c}
    1 &  a & 2 & a \
    0 &  1 & 0 & frac{-a^2}{a^2-2} \
    0 & 3-a & -2 & 2-a 
    end{array}right) 
& xrightarrow{small
        begin{array}{r}
        mathbf{r}_1 rightarrow mathbf{r}_1 - mathbf{r}_2a \
        mathbf{r}_3 rightarrow mathbf{r}_3 - mathbf{r}_2(3-a) \
        end{array}} 
    left(begin{array}{ccc|>{displaystyle}c}
    1  &  0 & 2 & -frac{2a}{a^2-2}  \
    0 &  1 & 0 & frac{a^2}{a^2-2} \
    0 & 0 & -2 & frac{1}{a^2-2}(-a^2+2a-4)  
    end{array}right)  \[10pt]
left(begin{array}{rrr|>{displaystyle}r}
    1  &  0 & 2 & -frac{2a}{a^2-2}  \
    0 &  1 & 0 & frac{a^2}{a^2-2} \
    0 & 0 & -2 & frac{1}{a^2-2}(-a^2+2a-4)  
    end{array}right)
&  xrightarrow[hphantom{textstyle~mathbf{r}_3 rightarrow mathbf{r}_3 - mathbf{r}_2(3-a)}]%
      {small
        begin{array}{r}
        mathbf{r}_1 rightarrow mathbf{r}_1 + mathbf{r}_3 \
        mathbf{r}_3 rightarrow frac{mathbf{r}_3}{-2}
        end{array}}
left(begin{array}{rrr|>{displaystyle}r}
    1 &  0 & 0 & -frac{a^2+4}{a^2-2}  \
    0 &  1 & 0 & frac{a^2}{a^2-2} \
    0 &  0 & 1 & frac{frac{a^2}{2}-a+2}{a^2-2}
    end{array}right) 
end{align}

end{document}