On the difference between gauss-markov and least squares estimates

For the usual full rank univariate least squares regression model y = XB + e, E(e) = 0, E(ee) = A, the equality of the estimates occurs when B-B^* = (XA⁻¹X)⁻¹XA^-1y-(XX)⁻¹Xy = 0. A necessary and sufficient condition for this equality is that A has some N - k + 1 roots equal where N is the rank of A and k is the rank of X.