Future X-ray astrophysics experiments require multiplexed readout of high fill-factor, kilo-pixel arrays of transitionedge sensors (TESs), with very high spectral resolution over a broad range of energies. In this paper we report on a prototype kilo-pixel array of Mo/Au TESs readout with 8-column by 32-row time-division multiplexing (TDM). This system is being used to demonstrate the critical detector and readout technology for ESA's Athena X-IFU, and when complete will be used in laboratory astrophysics experiments. Our array and TDM readout have demonstrated a combined full-width-at-half-maximum energy resolution, including > 200 pixels, of: 1.95 eV for Ti-Kα (4.5 keV), 1.97 eV for Mn-Kα (5.9 keV), 2.16 eV for Co-Kα (6.9 keV), 2.33 eV for Cu-Kα (8 keV), 3.26 eV for Br-Kα (11.9 keV). The 1 sigma statistical errors are ≤0.01 eV for all spectra. These results meet the broad-band resolution requirements for X-IFU with margin.