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FM Matrix Applications

1.8 Matrix Applications: Solving Systems of Equations

Representing Systems of Equations in Matrix Form

  • Matrices provide a concise way of representing systems of linear equations (i.e. multiple related linear equations). The equations are represented in 3 matrices:
    • A square coefficient matrix, generally denoted by a capital A, where each column lists the coefficients of a corresponding variable, while each row corresponds to a different equation.
    • A variable column matrix which lists the variables. It is denoted by a capital X. This is multiplied by the coefficient matrix to form the left-hand side of the matrix equation.
    • A column matrix of constants, which lists the constant in each equation. It is denoted by a lower-case b. This forms the right-hand side of the matrix equation.
    • Equation for a system of linear equations in matrix form:


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1.7 Matrix Applications: Dominance Matrices

Dominance Matrices

  • A dominance matrix is a binary matrix which has individuals listed along the rows and columns. A 1 element indicates the individual corresponding to the row is “dominant to” or “beat” the individual corresponding to the column. A 0 element indicates this is not the case.
  • This type of matrix is useful when modelling systems such as round robin competitions, where multiple people play against each other in rounds, with each round producing a winner.
  • This type of matrix is also known as a one-step dominance matrix.
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1.6 Matrix Applications: Communication Matrices

Communication Matrices

  • Communication matrices provide a numerical method for visualising communication networks. Each individual in the network is listed along the rows and columns of the matrix.
  • A “1” element indicates the sender corresponding to the row can communicate directly to the receiver corresponding to the column. A “0” means they cannot.
  • By convention all elements on the main diagonal are “0”s, as it is generally redundant to analyse an individual’s ability to communicate with itself.
  • A basic communication matrix is also referred to as a one-step communication matrix as it shows the communication allowed via a single “step” (i.e. direct communication).
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1.5 Binary and Permutation Matrices

Applications of Binary Matrices

Note: if you cannot remember what a binary matrix is, revise notes for 1.1 Matrices: Definition and Unique Cases.

  • Binary matrices are used to represent correlation between datasets. If two categories are correlated, a 1 is placed in the element corresponding to the two, if they are not, a 0 is placed instead.
  • They are also used to represent dominance. If one category is “greater than” another, a 1 is placed in the corresponding element, else a 0 is placed instead.

Note: we will further explore dominance matrices in notes 1.7 Matrix Applications: Dominance Matrices.

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1.4 Basic Matrix Applications

Using Matrices to Present Information

  • Matrices can be used to present information in a tabular form.
  • Similar to 2-way frequency tables, two categorical datasets are listed as the rows and columns, respectively, and each element represents a numerical value corresponding to the category in its row and column.
  • When presenting data in a matrix, the categories should be listed at the top of each column, and to the left of each row.
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