documentclass[11pt]{book}

usepackage{amsmath,amsfonts,amssymb,amsthm}

makeatletter
renewcommand*env@matrix[1][*c@MaxMatrixCols c]{%
  hskip -arraycolsep
  let@ifnextcharnew@ifnextchar
  array{#1}}
makeatother

usepackage{tikz}

begin{document}

begin{equation}
begin{bmatrix}[cccc|c]
  1 & 0 & -1 & 0 & 0 \
  0 & 1 & 2 & 0 & 0 \
  0 & 0 & 0 & 1 & 0
end{bmatrix}\
end{equation}

end{document}