Basically, “matrix” management structures are contrasted with more traditional “line” structures. A familiar synonym to the “line” might be the “silo,” where there is a clear reporting structure and every management level neatly nests into the next. There is a clear chain of command, which is good, but interaction between lines/silos can be a bit cumbersome.
Matrix management emerged in the 70s as a way to break down these silos. In the matrix, an employee reports not only to the line manager (the employee’s functional boss), but to a project manager (the employee’s operational boss). I suppose in the parks system, this could mean you might have a flying cleanup crew who reports to the the cleaning department manager (functional) as well as the local park supervisor (operational).
If done well, this system has the potential to be more flexible and responsive. If done poorly, a matrix structure can get bogged down with top-heavy, competing management layers and conflicting loyalties among confused employees. It seems to me that this system can only work if the line management has respect for the operational mission (ie: delivering services to people), and if there is a way to tell how well the system is working. This system makes more sense in companies that deal with new projects year after year (developing a new product, creating a specific promotional campaign, etc.), requiring new teams and fresh thinking. It doesn’t seem to make much sense in a local park, whose features and users do not actually change frequently. I am especially not sure how moving around local park supervisors would do anything but add to the confusion, and diminish respect for the horizontal, operational layer of the matrix (ie: the folks who actually run the local park).
If you want more info about matrix management, here is a good article to start with (from the Harvard Business Review, critical of matrix structures), you are interested. Here is another and another. And here is a blog dedicated to matrix management.