A system-on-chip (SoC) photonic-electronic linear-algebra accelerator with the features of wavelength-division-multiplexing (WDM) based broadband photodetections and high-dimensional matrix-inversion operations fabricated in advanced monolithic silicon-photonics (M-SiPh) semiconductor process technology is proposed to achieve substantial leaps in computation density and energy efficiency, including realistic considerations of energy/area overhead due to electronic/photonic on-chip conversions, integrations, and calibrations through holistic co-design methodologies to support linear-detection based massive multiple-input multiple-output (MIMO) decoding technology requiring the inversion of channel matrices and other emergent applications limited by linear-algebra computation capacities.