random vector expectations

I come across a question about calculating the expectation for random vectors

and I have no idea how to do with it.

Anyone could give me a hand? Thanks for your attention.

The PDF of , conditioned on , is Gaussian with mean vector and covariance matrix

, i.e.,

,

and the PDF of and , given , are also both Gaussian:

.

.

We also have

.

are all known and is an identity matrix.

Now what is the conditional expectation of , given some

observed and , i.e.,

The answer is something like

But don't know how