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\begin{equation}
\begin{bmatrix}
    f_1 & g_1 & 0 & \cdots & 0\\
    e_2 & f_2 & g_2 & \ddots & \vdots\\
    0 & e_3 & \ddots & \ddots & 0\\
    \vdots & \ddots & \ddots & \ddots & g_{J-2}\\
    0 & \cdots & 0 & e_{J-1} & f_{J-1}
\end{bmatrix}
\begin{pmatrix}
\vphantom{f_1}\Psi_1^{n+1}\\
\vphantom{f_2}\Psi_2^{n+1}\\
\vphantom{0}\vdots\\
\vphantom{\vdots}\Psi_{J-2}^{n+1}\\
\vphantom{0}\Psi_{J-1}^{n+1}
\end{pmatrix} = 
\begin{pmatrix}
r_1^n\\
r_2^n\\
\vdots\\
r_{J-2}^n\\
r_{J-1}^n
\end{pmatrix}.
\end{equation}

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}

\usepackage{mathtools}

\usepackage{lmodern}
\usepackage{microtype}

\begin{document}
  \begin{equation} 
    \begin{bmatrix} 
        f_1 & g_1 & 0 & \cdots & 0\\ 
        e_2 & f_2 & g_2 & \ddots & \vdots\\ 
        0 & e_3 & \ddots & \ddots & 0\\ 
        \vdots & \ddots & \ddots & \ddots & g_{J-2}\\ 
        0 & \cdots & 0 & e_{J-1} & f_{J-1} 
    \end{bmatrix}
    \renewcommand*{\arraystretch}{1.25}
    \begin{pmatrix} 
      \Psi_1^{n+1}\\ 
      \Psi_2^{n+1}\\ 
      \vdots\\
      \Psi_{J-2}^{n+1}\\ 
      \Psi_{J-1}^{n+1}
    \end{pmatrix} = 
    \begin{pmatrix} 
      r_1^n\\ 
      r_2^n\\ 
      \vdots\\ 
      r_{J-2}^n\\ 
      r_{J-1}^n 
    \end{pmatrix}. 
  \end{equation}
\end{document}