Variance-Covariance Matrix Calculation with constraints, error propagation

I'm trying to propagate errors.
The function is F= sum_i A_i*B_i
Each element of A,B has associated errors, the errors on B are uncorrelated, but the errors on A are correlated by the constraint
sum_i A_i =1.

I can work out the off diagonal terms of the variance-covariance matrix for the case when A & B have 2,3 elements. But I can't work it out from greater number of elements.
Is this possible? and if so how?