Variance calc with weighted coefficients

I want to understand the calculation of the variance of a sum of rvs where each variable is weighted using a constant coefficient. I post the full question and answer here and I highlight the areas that I am struggling to follow.

Question.

is a sample from a population with mean and variance .

The sample is not random, for . Let .

(a) Give the condition on a constants for U to be unbiased estimator of .

(b) Under this condition, calculate MSE(U).

Answer.

(a) is not a problem, just need to calculate a if which gives

and is the condition.

(b) given (a) is met, then MSE(U)=Var(U):

(1)

*and starting from (2) I find it tricky to follow the solution:*

(2)

(3)

It looks to me like a sum of all entries of the variance matrix of an nx1 vector... but I cannot yet move beyond that.

Then it gets OK again, a straightforward substitution

(4).

I'd appreciate a bit of clarification on manipulations in (2) and (3)...