What the Hell is... Matrix Organization?
A matrix organization takes workers with matching skill sets and places them together, allowing a project manager to select the employees that are needed for a given task. In this decentralized system, everything flows through a project manager, who then reports to a corporate executive or department head.
Matrix organizations arose in response to "silos," divisions wherein a strict hierarchical structure left employees in isolated groups only responding and communicating to their direct supervisor. Since information was not effectively shared between the silos, project coordination fell to a C-Suite executive.
In a matrix organization, synergies can be realized by combining elements of project and functional management, as the sharing of resources and employees leads to cost reductions and a more efficient organization. The silos, which in a matrix organization consist of a group of programmers or engineers for example, are now required to communicate.
Critics question the sustainability of matrix organizations over time as the fluid nature of the corporate structure means that employees can have several superiors and talented workers might feel overburdened.
Origin:
The concept of the matrix organization rose to prominence in the 1970's and 1980's. It was outlined in Professor Jay R. Galbraith's work, Matrix Organization Designs: How to combine functional and project forms.

A matrix is not just about project management, multiple reporting lines can be used to create leadership by function, geography, product group etc...
Project management can be important in a matrix but can also be used as a prop for weak "lateral" management.
The structure itself just reflects the external and internal complexity of an organization, in itself it solves nothng - it is how people operate the matrix that makes the differnce.
See much more at www.lifeinamatrix.com or in my book Speed Lead - faster, simpler ways to manage people, projects and teams in complex companies