# Is the expectation of the inverse of a random matrix that has diagonal expectation also diagonal?

Suppose we have a symmetric positive definite random matrix $$\mathbf{A}$$ that has a diagonal matrix expectation, $$\mathbf{E}(\mathbf{A}) = \text{diag}(\mathbf{x}),$$ [like e.g. the Wishart distribution with diagonal scale matrix, $$\mathcal{W}(\text{diag}(\mathbf{x}),n)$$].

Is it true that $$\mathbf{E}(\mathbf{A}^{-k}),$$ where $$k$$ is an arbitrary positive integer is also a diagonal matrix?

Edit: For the Wishart this is true. I am interested in the general statement.