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Definition: a Matrix Organization is an organizational structure that is matrix-shaped. It has 2 axes, rather than a pyramid-shape such as is the case in a traditional organizational hierarchy. Besides the vertical authority, a MO has some form of horizontal or lateral authority, influence, or communication.
For employees this means that they must deal with dual lines of authority, dual sources of reward and punishment, dual reporting and communication channels, shared responsibility and accountability.
MOs require extensive and very effective communication systems and can result in higher overhead because they create more management positions.