Centering column in matrix equations

Question

I'm looking for a way to centre-align the centre column (\xrightarrows) in this construction.

\documentclass{article}
\usepackage{mathtools}
\begin{document}
\begin{alignat}[2]
    \begin{bmatrix*}[r]
        2 & 2 & \vline & 3 \\
        1 & -3 & \vline & 2
    \end{bmatrix*} & \xrightarrow{-R_2 \to R_1} && \begin{bmatrix*}[r]
        1 & 5 & \vline & 1 \\
        1 & -3 & \vline & 2
    \end{bmatrix*} \\
    \begin{bmatrix*}[r]
        1 & 5 & \vline & 1 \\
        1 & -3 & \vline & 2
    \end{bmatrix*} & \xrightarrow{3 R_1} && \begin{bmatrix*}[r]
        3 & 15 & \vline & \mathllap{-}1 \\
        1 & -3 & \vline & 2
    \end{bmatrix*}
\end{alignat}
\end{document}

Previous answers I've seen have used:

• an array environment to align columns (Centering equations within alignat command), but in this case the rows become too close together and can't be separated by adjusting the spacing of array rows because this also affects the matrix environments.
• an alignat environment (Centering column in alignat environment?) but this makes the centered column a text environment. I need a math environment.

Answer 1

The alignat environment works just fine here. What made you believe it turns the centre column in textmode? Note the starred versions of the matrix environmùents require mathtools. I took the opportunity to align the right brackets of the righmots matrices, using the \mathllap command defined by this package:

\documentclass{article}
\usepackage{mathtools}

\begin{document}

\begin{alignat*}{2}
    \begin{bmatrix*}[r]
        2 & 2 & \vline & 3 \\
        1 & -3 & \vline & 2
    \end{bmatrix*} & \xrightarrow{-R_2 \to R_1} && \begin{bmatrix*}[r]
        1 & 5 & \vline & 1 \\
        1 & -3 & \vline & 2
    \end{bmatrix*} \\[1ex]
    \begin{bmatrix*}[r]
        1 & 5 & \vline & 1 \\
        1 & -3 & \vline & 2
    \end{bmatrix*} & \xrightarrow{-R_2 \to R_1} && \begin{bmatrix*}[r]
        0 & 8 & \vline & \mathllap{-}1 \\
        1 & -3 & \vline & 2
    \end{bmatrix*}
\end{alignat*}

\end{document} 

To centre the \rightarrows in their column, we can use the eqparbox package. I defines a \eqmathbox command, which types its contents in math mode and uss a system of tags. All boxes sharing the same tag have the width of the wides of their contents, which is centred in its box:

\documentclass{article}
\usepackage{mathtools}
\usepackage{eqparbox}
\newcommand{\eqmathbox}[2][M]{\eqmakebox[#1]{$\displaystyle#2$}}

\begin{document}

\begin{alignat*}{2}
    \begin{bmatrix*}[r]
        2 & 2 & \vline & 3 \\
        1 & -3 & \vline & 2
    \end{bmatrix*} & \eqmathbox{\xrightarrow{-R₂ + R_1\to R₁}} && \begin{bmatrix*}[r]
        1 & 5 & \vline & 1 \\
        1 & -3 & \vline & 2
    \end{bmatrix*} \\[1ex]
    \begin{bmatrix*}[r]
        1 & 5 & \vline & 1 \\
        1 & -3 & \vline & 2
    \end{bmatrix*} & \eqmathbox{\xrightarrow{-R₂ \to R₁}} && \begin{bmatrix*}[r]
        0 & 8 & \vline & \mathllap{-}1 \\
        1 & -3 & \vline & 2
    \end{bmatrix*}
\end{alignat*}

\end{document}