The square (n=number of rows=number of columns) matrix can be used for analysis of interdependencies (among plans, documents, tasks, goals, or any other activities or artifacts) by entering marks (for example, "X") in the matrix as follows:

  • All the cells in the diagonal are marked, meaning that each activity (or artifact) feeds itself.
  • A mark is entered in cell (x,y) when the column activity "y" must precede the row activity "x".
  • The precedence relation can be ["x must end before y can start"] or there can be overlap ["x must start before y can start"].
  • Numbers can be entered to reflect the criticality of the precedence relations (e.g., 1 to 9, where 1 means the precedence relation is barely critical and 9 means the precedence relation is highly critical).
  • The only mathematical restriction is that the number of rows and columns must be the same (i.e., the matrix must be nxn).
  • In practice, however, this kind of analysis is practical when 550, the analysist's mind is taxed, and when n>100 the analyst's mind starts melting down.
    • In this cases, the total analysis must be conducted at several levels of aggregation. This entails a decomposition of activities (or artifacts) such that an entry in the top level matrix is decomposed into a number of entries for lower level matrix. Existing software can support the decomposition process to any number of levels, but the analyst's mind again becomes the limiting factor if the number of levels is greater than 3 (4 or 5 at the most). If the analysis requires 5,000 entries in a hierarchy of 16 levels of matrices, the problem is that the real-world problem is not understood.


The UN Millennium Development Goals (MDGs) are the following:

1. Eradicate extreme poverty and hunger
2. Achieve universal primary education
3. Promote gender equality and empower women
4. Reduce child mortality
5. Improve maternal health
6. Combat HIV/AIDS and other diseases
7. Ensure environmental sustainability
8. Develop a global partnership for development

It is self-evident that these goals are not independent of each other. Analysis of the goals readily show that they are not amenable to structuring as a precedence diagram. However, the complex open-loop and closed-loop interactions can be sorted out using the square matrix triangularization technique. The result is a block triangular matrix that shows which goals can be serialized and which goals are so tightly coupled that they need to be worked together.

Precedence Scenarios

  • Scenario 1 assumes that attaining MDG3 (gender equity) must precede all the other goals.
  • Scenario 2 assumes that none of the MDGs is a critical precondition to all the others.

Partitioned Matrices for Scenarios 1 and 2

The partitioned matrices for scenarios 1 and 2 are shown below.


See Problematics for the software used.

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The following are some recommended online resources: