Standard Error

Below are two similar problems.

For the question above I can do i and know how to do iii but don't see how to get the standard error of eta. The answers are all below. It's just the working out I don't get.

For the question above, the same problem arises. It's a similar question. I can do i, ii, and know how to do iv, but don't know how to find the standard error of eta (although I can find the estimate of eta).

I think it has to do with the formula \(var=\sigma^2(A^{T}A)^{-1}\). Looking at the answers below, they take the square root of S^2 times something that contains the numbers in the \(A^{T}A\) matrix, which is that matrix shown in (a)i answers.