Multiplying matrices of matrices

A question about multiplying the matrix below that is a variation Henderson's mixed model equations:

|M1  M2| |M5  M6|
|M3  M4| |M7  M8|


whre M1-M8 are matrices. I'm not exactly sure what the result should look like. I think it should result in a single matrix as I need to take the trace of the result. Feedback appreciated.

EDIT: I don't know what happened to my edits from last night. Here's an example of two matrices: http://escholarship.org/uc/item/9wc564c1#page-34 (Eq 2.2). Maybe the best approach is to consider the left and right matrix as m*(p+q)X*m*(p+q) matrices where X is is nXp, Z is nXq and m is the number of clusters. Then you don't have to worry about the subcomponents. It's simple (but very large in some cases) crossproducts of two matrices. Sound right?

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Welcome to Math.SE! This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. –  Joe Tait Feb 25 '14 at 22:15
@Joe Tait-I put some clarifying information on this last night. I was at a different computer and could not get logged in, so maybe the changes went to moderation. Anyway, I don't seem to be able to post images here like at Crossvalidated and your help does not show how to format side-by-side matrices, so I left a link that shows for what I'm aiming. If the changes are not posted in the next couple of hours, I'll try editing this post once again. –  William Shakespeare Feb 26 '14 at 13:18