Complex Wishart matrices are a key part of many multidimensional signal processing algorithms, and expectations of expressions containing them are needed to quantify their performance. In this paper we derive concise formulas for E{WAW}, E{W-1AW-1}, and other useful expressions, where W follows the complex Wishart distribution and A is a deterministic matrix. A need not be symmetric. We show how the results can be used to quantify the performance of adaptive array processing algorithms.