Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 13J / b / Solution

Since $Y-HY=(I-H)\mu +(I-H)\epsilon$, the cross term has mean zero and $H$ is symmetric. Thus the expectation is $\Vert (I-H)\mu {\Vert }^2+{\sigma }^2\mathrm{tr}(I-H{)}^2$, which expands to the stated expression.