Copying and editing David Carlisle's code from Vertical dashed line of a custom height yields
\documentclass{article}
\usepackage{amsmath}
\usepackage{arydshln}
\makeatletter
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
\hskip -\arraycolsep
\let\@ifnextchar\new@ifnextchar
\array{#1}}
\makeatother
\begin{document}
\begin{equation*}
\begin{bmatrix}[cccc:c]
a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\
a_{21} & a_{22} & \cdots & a_{2n} & b_2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn} & b_n
\end{bmatrix}
\end{equation*}
\end{document}
EDIT
Since OP's intended output is different from what's in his/her image, here's some code that does both.
\documentclass{article}
\usepackage{amsmath}
\usepackage{arydshln}
\makeatletter
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
\hskip -\arraycolsep
\let\@ifnextchar\new@ifnextchar
\array{#1}}
\makeatother
\begin{document}
\begin{equation*}
\begin{bmatrix}[cccc:c]
a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\
a_{21} & a_{22} & \cdots & a_{2n} & b_2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn} & b_n
\end{bmatrix}
\end{equation*}
Without the dashes shown in OP's question
\begin{equation*}
\left[
\begin{matrix} \\ v_1 \\ \\\end{matrix}
\Biggm\vert
\begin{matrix} \\ v_2 \\ \\\end{matrix}
\right]
\end{equation*}
\end{document}
\phantom{.}at the end of line 7 (the last line with the three consecutive ampersands). – diabonas Sep 20 '17 at 12:36