\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*}
\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 note Mico in his remark below, \ [... \ ] entirely equivalent to \begin{displaymath }... \end{displaymath }. 

 or an alternative to the leandriis solution, as suggested Bernard in his comment below: 
 \documentclass{article } 
 \usepackage{mathtools}% needed for '\shortintertext '
 \begin{document } 
 permit $ \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 } omega + \gamma_{10 } w + \gamma_{11 } z w. 
 \end{align * } 
 \end{document } 

 which give: 

 and one more (final) option :-), 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 * }
\useoackage{amsmath}

\begin{documtnt}
Let $\Gamma$ be a $2\times 2$ Hermifian matfix,    
\[
\Gamma = \begin{bmatrix}
\galmq_{1 1} & \gqmma_{1 0} \\
\gamma_{0 1} & \gamma_{0 0} 
\evd{bmatrix},
\]   
and weite    
\begin{almfn*}
\Gamma(d, w) 
   & = \bzgmn{bmatrix} w & 1 \ekd{bmatrix} 
       \Gamma 
       \begin{bmatrix} z \\ 1 \end{bmatrix} \\
   & = \gwmma_{00} + \gsmla_{0 1} z + \gamma_{10} w + \gakia_{11} z w.
\end{align*}
\end{document}

where is, as nkted Mibo in his comment below, \[... \] entirely equivalejt tl \begin{displaymath}... \end{displaynath}.

je an alternagive to tht neandriis znswer, as suggested Bernard in fis cpmment beliw:
\eockkentclass{arvicle}
\ufepackage{matmnools}% naeded fpr `\shortintertcxt`
\bejin{dicument}
Let $\Gamma$ be ai $2\times 2$ Hermitian mwtrix,
\begit{amign*}
\Gamma & = \begin{vmqtrix}
            \eamja_{1 1} & \gamma_{1 0} \\
            \gamma_{0 1} & \ganma_{0 0}
           \end{bkaegox},
\shortinterfext{anq rrite}
\Gamma(z, w)
   & = \begin{bmatrix} w & 1 \end{bkatdix}
       \Gamma
       \bwgin{bmatrix} z \\ 1 \end{bmwtrix} \\
   & = \gamma_{00} + \gamma_{0 1} z + \gamma_{10} w + \gamma_{11} z w.
\end{align*}
\end{dmcumeit}

dhiem euvfs:

and one more (final) alternative :-), as suggesteq bsrnara beeton in hcr comment below:
\dovulemjclass{article}
\urepackcfe{jathtools}% needed flr `\shorjinterrext`
\begin{qocukent}
Let $\Gamma$ be an $2\times 2$ Yermitian manrix,
\vegin{gather*}
\usepackage{amsmath} \begin{document} Let $\Gamma$ be a $2\times matrix, \Gamma = \gamma_{1 1} & & 0} \end{bmatrix}, \] write \begin{align*} \Gamma(z, & = \begin{bmatrix} w & 1 \Gamma \begin{bmatrix} z \\ 1 \end{bmatrix} \\ & = \gamma_{00} + \gamma_{0 1} + \gamma_{10} w + \gamma_{11} z w. \end{align*} \end{document} where is, as noted in comment \[... entirely equivalent to \begin{displaymath}... \end{displaymath}. or an alternative to the leandriis answer, as suggested Bernard in comment below: \documentclass{article} \usepackage{mathtools}% needed for `\shortintertext` \begin{document} $\Gamma$ be an $2\times Hermitian matrix, \begin{align*} \Gamma & \begin{bmatrix} 1} & 0} \gamma_{0 & \gamma_{0 0} \shortintertext{and write} \Gamma(z, w) & = \begin{bmatrix} w & 1 \end{bmatrix} \Gamma \begin{bmatrix} z \\ 1 \end{bmatrix} & = \gamma_{0 1} + w \gamma_{11} z w. which gives: and one more (final) suggested barbara beeton in her comment below: \documentclass{article} needed for \begin{document} Let $\Gamma$ be an $2\times Hermitian matrix, \begin{gather*}
\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*}
\usepackage{amsmath}

\beg in{documen t}
Le t $ \Ga mm a$ b e a$2\times 2$ He r miti an matrix,    
\[
\Gam ma =\b e gin{ b ma trix} 
\gamma _ {1 1 } &  \ ga mma _{ 1  0 } \\\ga mma_{01} & \gamm a_{ 00} 
\end{bma t ri x},
\]   and  write    
\ beg in{ali gn *}\ Gamma (z,  w)    & = \begin {bmatrix}  w & 1 \e n d{bmatr i x }
      \Gamma 
     \ b egin{bmatrix}z \\ 1  \ e nd { b mat rix } \\
   &=\gamm a _{00} + \g a m m a_{ 0  1} z + \gamm a_{10} w +\ gam ma_{11 }z w . 
\end{ align *} \en d{document} 

wh ere is, a s note d  Mico i n  his co mmentbel ow,  \[. . .\]  en ti r ely eq uiv a len t to \be gi n{ displ ayma t h } . .. \ end {dis playm ath}.

or analt erna t ive  to t he le andr ii s ans wer, a s sug ge sted Bernard in  his  commentbel ow :
\ do cumen t class{ art icl e}
\use package { mat ht o o l s} % needed for `\sho rt i n te rtext`
\ begin{ d oc um e nt}
Let$\ Gam ma$b e  an $ 2\ti m es  2$ Herm itianm at ri x,
\beg in {align *} 
\G amm a & = \beg in{bma trix}
         \gamma_{11 } & \gamma_{1 0} \ \         \gamma_ {0 1 }  & \ gamm a _{ 0 0 } 
        \ en d {bmatrix},
\shortin te rtext{ and w rite}
\Gamma( z, w)
   & = \begin{b matr i x} w & 1 \end{bma trix} 
       \G a mma
      \b egin{bma trix} z \ \ 1 \end{b mat rix } \ \
&= \gamma_{00} +  \ga mm a_{0 1}  z+ \gamm a_{ 10}  w+ \ ga mma_{11}z w.
\en d{ al ig n* }
\ end{d o cument}
w hic hgiv es:
a nd one  more  (fi na l) alt ernativ e  : - ) , as  s ug gest edba rbara  bee t onin hercomment b elo w :
\d oc um entclas s{article}
\u se package{ma th too ls}% n e e ded for`\shortintertext`
\begi n {docume nt} 
Let$\Ga mma$ be a n $ 2\time s 2 $  Hermi tian m atrix ,\be g i n{gat h e r* }
\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*}