Q:

How to write matrix with math symbols in equation?

Here is my try
Let $\Gamma$ be an $2\times 2$ Hermitian matrix,\\
\begin{displaymath}
\begin{equation}
$\Gamma$ = $\begin{bmatrix} $\gamma 1$ &  $\gamma 2$\\ $\gamma 3$ &$\gamma 4$
\end{bmatrix}
\end{equation}
\end{displaymath}

A:

let me convert my comment to an answer:

your equation is written on very wrong way:

it is not allowed to nest \begin{equation} ... \end{equation} inside \begin{displaymath} ... \end{displaymath}, 
also use $ inside equation is not allowed (it is intend for use for math expression in text)

your equation can be written on many ways, besides shown in other answers:

\documentclass{article}
\usepackage{amsmath}

\begin{document}
Let $\Gamma$ be a $2\times 2$ Hermitian matrix,    
\[
\Gamma = \begin{bmatrix}
\gamma_{1 1} & \gamma_{1 0} \\
\gamma_{0 1} & \gamma_{0 0} 
\end{bmatrix},
\]   
and write    
\begin{align*}
\Gamma(z, w) 
   & = \begin{bmatrix} w & 1 \end{bmatrix} 
       \Gamma 
       \begin{bmatrix} z \\ 1 \end{bmatrix} \\
   & = \gamma_{00} + \gamma_{0 1} z + \gamma_{10} w + \gamma_{11} z w.
\end{align*}
\end{document}

where is, as noted Mico in his comment below, \[ ... \] entirely equivalent to \begin{displaymath} ... \end{displaymath}.

or an alternative to the leandriis answer, as suggested Bernard in his comment below:
\documentclass{article}
\usepackage{mathtools}% needed for `\shortintertext`
\begin{document}
Let $\Gamma$ be an $2\times 2$ Hermitian matrix,
\begin{align*}
\Gamma & = \begin{bmatrix}
            \gamma_{1 1} & \gamma_{1 0} \\
            \gamma_{0 1} & \gamma_{0 0}
           \end{bmatrix},
\shortintertext{and write}
\Gamma(z, w)
   & = \begin{bmatrix} w & 1 \end{bmatrix}
       \Gamma
       \begin{bmatrix} z \\ 1 \end{bmatrix} \\
   & = \gamma_{00} + \gamma_{0 1} z + \gamma_{10} w + \gamma_{11} z w.
\end{align*}
\end{document}

which gives:

and one more (final) alternative :-), as suggested barbara beeton in her comment below:
\documentclass{article}
\usepackage{mathtools}% needed for `\shortintertext`
\begin{document}
Let $\Gamma$ be an $2\times 2$ Hermitian matrix,
\begin{gather*}
\Gamma = \begin{bmatrix}
            \gamma_{1 1} & \gamma_{1 0} \\
            \gamma_{0 1} & \gamma_{0 0}
         \end{bmatrix},
\shortintertext{and write}
\begin{split}
\Gamma(z, w)
   & = \begin{bmatrix} w & 1 \end{bmatrix}
       \Gamma
       \begin{bmatrix} z \\ 1 \end{bmatrix} \\
   & = \gamma_{00} + \gamma_{0 1} z + \gamma_{10} w + \gamma_{11} z w.
\end{split}
\end{gather*}
\end{document}

which gives

A:

Some suggestions:

The contents of an environment such as displaymath are in math mode automatically -- no need to encase math elements in $ ...$. 
Use an align* environment for multi-line unnumbered equations.

\documentclass{article}
\usepackage{amsmath} % for 'bmatrix' and 'align*' environments
\begin{document}
Let $\Gamma$ be a $2\times2$ Hermitian matrix,
\begin{displaymath}
\Gamma = \begin{bmatrix} 
            \gamma_{11} & \gamma_{10} \\ 
            \gamma_{01} & \gamma_{00}
         \end{bmatrix}\,,
\end{displaymath}
and write
\begin{align*}
\Gamma(z,w) &= \begin{bmatrix} w &  1 \end{bmatrix} 
               \Gamma
               \begin{bmatrix} z \\ 1 \end{bmatrix} \\
            &= \gamma_{00}+\gamma_{01}z+\gamma_{10}w+\gamma_{11}zw.    
\end{align*}
\end{document} 

A:

A variant using a single align* environment and intertext:
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
\intertext{Let $\Gamma$ be an $2\times 2$ Hermitian matrix,}
\Gamma =& \begin{bmatrix} \gamma_{11} &  \gamma_{10}\\ \gamma_{01} &\gamma_{11}\end{bmatrix},
\intertext{and write}
\Gamma(z,w) =& \begin{bmatrix} w & 1 \end{bmatrix} \Gamma \begin{bmatrix} z \\ 1 \end{bmatrix}\\
=& \gamma_{00}+\gamma_{01}z+\gamma_{10}w+\gamma_{11}zw. 
\end{align*}
\end{document}

