Radiative Forcing of climate by sulfate aerosols is thought to exert a direct shortwave forcing of climate comparable in magnitude but opposite in sign to the longwave forcing by anthropogenic greenhouse gases. However, recent work has questioned the cooling influence of aerosols because of possible influences of absorbing aerosol that might offset the negative forcing of a purely scattering aerosol. Here we examine the optical and radiative properties of sulfate aerosol internally and homogeneously mixed with an absorbing substance and evaluate the dependence of forcing on controlling variables including the imaginary component of the index of refraction, particle radius, solar zenith angle, and surface reflectance. For example, for a mixture of sulfate aerosol and soot that gives rise to an imaginary component of the index of refraction equal to 0.008 (corresponding to a single scattering albedo, w = 0.9 for particle radius, r = 120 nm with surface albedo, R = 0.15) the normalized global-average (1/2 the quantity over the illuminated hemisphere) forcing (forcing divided by sulfate column burden) is -575 W/g(SO4) compared to -800 W/g(SO4) for pure sulfate aerosol (imaginary component equal to zero). We also map out the boundaries for which the forcing is negative (cooling) versus positive (warming). For example, for r = 120 nm, and R = 0.15 the normalized global-average forcing is positive for imaginary component greater than about 0.04 (w < 0.9). Instantaneous normalized forcing becomes positive at even smaller values of imaginary index (imaginary component, 0.008; single scattering albedo, 0.9) for the sun near zenith. Comparison with single scattering albedos measured at the ARM SGP site and elsewhere suggest that the sign of tropospheric aerosol forcing is predominantly negative.