An explicit formula is derived to compute the $A$-optimal design weights on linearly independent regression vectors, for the mean parameters in a linear model with homoscedastic variances. The formula emerges as a special case of a general result which holds for a wide class of optimality criteria. There are close links to iterative algorithms for computing optimal weights.
"Optimal Weights for Experimental Designs on Linearly Independent Support Points." Ann. Statist. 19 (3) 1614 - 1625, September, 1991. https://doi.org/10.1214/aos/1176348265