documentclass{article}
usepackage{amsmath}
usepackage{mleftright}

begin{document}
[
renewcommandarraystretch{1.3}
A_{ij} = mleft[
begin{array}{c|c|c|c|c}
     A_{c,rho} & A_{c,u_r} & A_{c,u_{theta}} & A_{c,u_z} & A_{c,T}     \ hline
     A_{r,rho} & A_{r,u_r} & A_{r,u_{theta}} & A_{r,u_z} & A_{r,T}    \ hline
     A_{theta,rho} & A_{theta,u_r} & A_{theta,u_{theta}} & A_{theta,u_z} & A_{u_{theta},T}   \ hline
     A_{z,rho} & A_{z,u_r} & A_{z,u_{theta}} & A_{z,u_z} & A_{z,T}   \ hline
     A_{e,rho} & A_{e,u_r} & A_{e,u_{theta}} & A_{e,u_z} & A_{e,T}
end{array}
mright]
]

end{document}