Stack of matrices with unequal sizes

I have to write a list of two matrix equations. They are almost identical, but the elements in one matrix are slightly larger than the other one, which makes it look badly.

\begin{align}
\begin{pmatrix}
-\id & \id \\
\id & -\id \\
\end{pmatrix}  u = 0 \\
\begin{pmatrix}
\id &   \id \\
\id &  \id
\end{pmatrix}  v = 0
\end{align}


Is there a simple and elegant way to make the two matrices match size? I've found this, but it is a much complex case than mine. Is there something simpler?

I propose two versions, the second one is what I'd prefer.

The calc packages is necessary only for the first solution. As a general rule, never use align without alignment points.

\documentclass{article}
\usepackage{amsmath,calc}
\newcommand{\id}{\mathrm{id}}
\newcommand{\cid}{\makebox[\widthof{$-\id$}]{$\id$}}
\begin{document}
\begin{align}
\begin{pmatrix}
-\id & \id \\
\id & -\id \\
\end{pmatrix}  u &= 0 \\
\begin{pmatrix}
\cid &  \cid \\
\cid &  \cid
\end{pmatrix}  v &= 0
\end{align}
A different version
\begin{gather}
\begin{pmatrix}
-\id & \id \\
\id & -\id \\
\end{pmatrix}  u = 0 \\
\begin{pmatrix}
\id &  \id \\
\id &  \id
\end{pmatrix}  v = 0
\end{gather}

\end{document}


Here are two solutions, far from perfect. I use the variants of the matrix environments defined by mathtools, which allow to specify the alignment in a matrix and I modify the value of \arraycolsep when needed.

I also define a \nid math operator, which corrects the spacing between – and id in -\id, which is bad.

\documentclass{article}
\usepackage{textcomp} \usepackage{mathtools}
\usepackage[showframe]{geometry}
\DeclareMathOperator{\id}{{id}}
\newcommand*\nid{\ensuremath{-{\id}}}%
\newcommand*\varnid{\ensuremath{\mskip-5mu-{\id}}}
\newcommand*\varid{\ensuremath{\mskip-5mu\phantom{-}{\id}}}%

\begin{document}

\begin{align}
\begin{pmatrix*}[r]
\varnid & \id \\
\id & \nid \\
\end{pmatrix*} u = 0 \\
\setlength\arraycolsep{0.9em}
\begin{pmatrix*}[r]
\varid & \id \\
\id & \id
\end{pmatrix*} v = 0 \\
\setlength\arraycolsep{1.1em}
\begin{pmatrix*}[r]
\id & \id \\
\id & \id
\end{pmatrix*} v = 0
\end{align}

\end{document}