Many information quantities for multivariate normal distributions have nice closed forms. Their essential parts usually reduce to logs of minor determinant quotients so they combine nicely. Here's a list of them. All of them are somewhere in Adaptive Wireless Communications by Bliss and Govindasamy.
Notation
 [n]=\{1,\dots,n\}
 (x_n)_n ((x_n) for short) is a vector indexed by n
 (x_n)_{n\in S} (or (x_n)_S) is the subvector of (x_n)_n consisting only of elements from S
 If A is an n\times n matrix and S\subset [n] is nonempty, then A_S is the matrix formed by deleting all rows and columns of A in [n]\backslash S. Note in particular if A is positivedefinite then A_S is too.
 S is the number of elements in S.
All covariance matrices are assumed to be fullrank. By convention the determinant of an empty matrix is 1.
Real Identities

X \sim \mathcal{N}(0,\sigma^2), then:
h(X) = \frac{1}{2}\ln\left(2\pi e \sigma^2\right) 
(X_n) \sim \mathcal{N}(0,A),\ S\subset [n] then:
h\left( (X_n)_S \right) = \frac{1}{2}\ln\left((2\pi e)^{S} \operatorname{det}(A_S)\right) 
(X_n) \sim \mathcal{N}(0,A),\ S,T\subset [n] then:
h\left( (X_n)_S  (X_n)_T \right) = \frac{1}{2}\ln \left((2\pi e)^{S\cup TT} \frac{\operatorname{det}(A_{S\cup T})}{\operatorname{det}(A_T)}\right) 
(X_n) \sim \mathcal{N}(0,A),\ S,T\subset [n] and S\cap T = \varnothing then:
I\left( (X_n)_S ; (X_n)_T \right) = \frac{1}{2}\ln\left(\frac{\operatorname{det}(A_S)\operatorname{det}(A_T)}{\operatorname{det}(A_{S\cup T})}\right) 
(X_n) \sim \mathcal{N}(0,A),\ S,T,U\subset [n] and S\cap T=\varnothing then:
I\left( (X_n)_S ; (X_n)_T\middle(X_n)_U\right) = \frac{1}{2}\ln\left(\frac{\operatorname{det}(A_{S\cup U})\operatorname{det}(A_{T\cup U})}{\operatorname{det}(A_{S\cup T\cup U})\operatorname{det}(A_U)}\right)
Circularlysymmetric complex normal identities are identical except without the \frac{1}{2} factor.