We desinged a lattice-matched InGaAs/lnGaAsP quantum well laser that lases at $1.55{mu}m$ at room temperature, and calculated the gain spectrum as a function of injected carrier density and temperature. For the calculation of band structures and momentum matrix elements, we used a transfer JIlatrix method based on a block-diagonalized 8x8 second-order Ii$.$ P Hamiltonian. This lattice-matched quantum well lases in transverse electric mode. As the temperature increases, the lasing wavelength gets longer, the transparency carrier density increases, and the differential gain is reduced. The temperature dependence of the gain spectrum comes from the temperature dependence of the band structure and that of the Fermi function, and the latter contributes dominantly.nantly.