Matrix Diagram
The Matrix Diagram is an analysis tool that facilitates the systematic analysis of the strengths of
relationships between two or more sets of elements. It consists of a table whose main rows and
columns contain the elements being inter-related, with the rest of its cells containing symbols or
numbers that denote the strengths of relationship between the elements.
The elements being inter-related in a matrix diagram may be in the form of information, concepts,
conditions, activities, or other intangible items, as well as physical things such as people, equipment,
tools, and materials.
The matrix diagram can be used in almost all types of decision making that involves several options
or alternatives, or is affected by several factors. Examples of these include: 1) equal distribution of
major and minor assignments among members of a given project; 2) selection of a process,
equipment, or material for a given purpose; 3) identifying the most critical factors affecting a given
problem area; 4) matching of tasks to objectives, etc.
The elements belonging to the same row or column should have something in common, so that they
comprise a set that represents something. For instance, a matrix diagram that relates various
reliability tests to various failure mechanisms might show in its main row industry-standard reliability
tests and on its main column commonly-encountered failure mechanisms.
The strength of relationship between each reliability test and each failure mechanism may then be
denoted on the cell where they intersect with a symbol or a number (say, 1-3, with 3 denoting the
strongest relationship). Table 1 shows a simplified version of such a matrix diagram. This matrix
diagram shows, for instance, that if one wants to check the reliability of a set of samples with respect
to package cracking and ball lifting, then TCT should be the reliability test used instead of PCT or
HTOL.
Table 1. A Matrix Diagram Relating Reliability Tests to Failure Mechanisms
Package Cracking
Corrosion
Ball Lifting
Oxide Breakdown
TCT
PCT
HTOL
3
1
3
1
2
3
2
1
1
1
1
3
There are many types of matrices: 1) the L-shaped matrix; 2) the T-shaped matrix; 3) the Y-shaped
matrix; 4) the X-shaped matrix; and 5) the C-shaped matrix. The two most commonly used matrices,
however, are the L- and T-shaped matrices. The L-shaped matrix has a main row and a main
column that form an inverted 'L' to inter-relate two sets of items directly to each other, or a single set
of items to itself. The matrix shown in Figure 1 is an example of an L-shaped matrix.
On the other hand, the T-shaped matrix has its main column (or main row) separated in the middle
by a single main row (or single column) that appears in the middle of the matrix. The T-matrix is
used to inter-relate two sets of items (say, sets A and B) to a common third set of items (say, set C).
The items in set A will appear on the half of the main column above the main row, while those of set
B will be in the half below the main row. The items of the common set C will appear on the main row.
If the half-columns of sets A and B in the T-matrix described above are bent to allow inter-relation of
items of set A to those of set B, then a Y-shaped matrix results. Placing two T-shaped matrices
back-to-back, however, will result in an X-shaped matrix, which allows the inter-relation of four sets
of items to each other. Lastly, the C-shaped matrix is a 3-dimensional matrix that interrelates three
sets of elements simultaneously.
To construct a matrix diagram, the following steps are usually taken: 1) define the purpose of the
matrix diagram; 2) identify what sets of elements need to be included to meet the objective of the
matrix diagram; 3) assemble the best team that can inter-relate all the elements of the matrix; 4)
select the matrix format; 5) choose and define the relationship symbols; and 5) complete the matrix
diagram.
As an example, suppose that a supervisor wants to document the assigned tasks and expertise
levels of his engineers in matrix format. Since he needs to interrelate two different sets of
information (assignments and expertise levels) to a third common set (his engineers), then the Tshaped matrix is the best format for his purpose. Also, in this case, he has all the information he
needs to fill up the matrix, so no team is formed for the task. Had a more complex matrix been
required, then the right people must be called in to form the matrix. Figure 1 shows the T-matrix for
this example.
Figure 1. A T-shaped Matrix Diagram Defining the Assignments and
Expertise Levels of 3 Engineers
In the first half of the T-matrix above, graphical symbols (a circle and a triangle) were used to
interrelate the elements, with the circle denoting primary responsibility and the triangle denoting
secondary responsibility. The main reason for using graphical symbols in this portion is to have an
immediate visual indication of the distribution of the tasks among the engineers. One glance at the
table shows that the tasks were equally distributed.
In the second half of the T-matrix, numbers were used to denote the expertise levels of the
engineers. This is because there's a need to 'grade' the various expertise levels of the engineers.
Of course, symbols may also be used for this purpose, but doing so will also require an assignment
of a number to each symbol used. Lastly, using numbers in a matrix will allow mathematical
processing of the data (such as summing up the values of a row or column), which can be useful in
some cases.
The matrix diagram is a very versatile tool that can be used in many applications of the
manufacturing industry. Engineers who become 'matrix thinkers' gain the ability to conjure up matrix
diagrams whenever the need for it arises, allowing them to explore all available options
systematically before making a major decision.
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