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FM Matrices and their 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:

AX=b

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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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1.4 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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1.3 The Inverse of a Matrix

The Determinant of a 2×2 Matrix

  • The determinant of a 2 x 2 matrix can be calculated as:

\text { det }=A_{1,1} A_{2,2}-A_{1,2} A_{2,1}

  • The determinant of an identity matrix is equal to 1.

Example

A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]

The determinant of the above matrix is:

\operatorname{det}=1 * 4-2 * 3=-2

Requirements for a Matrix to be Invertible

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1.2 Elementary Matrix Operations

Addition and Subtraction

  • Matrix addition and subtraction is carried out on an element-by-element basis. The element C_{i,j} in the resultant matrix is found by carrying out addition or subtraction with the corresponding elements; A_{i,j} and B_{i,j} in the other matrices.
  • Matrices can only be added or subtracted from one another if they have the same dimensions.

Examples

\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right]+\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right]=\left[\begin{array}{ll} 2 & 2 \\ 1 & 1 \end{array}\right]

\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right]-\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right]=\left[\begin{array}{cc} 0 & 2 \\ -1 & 1 \end{array}\right]

Scalar Multiplication

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1.1 Matrices: Definition and Unique Cases

Matrices

  • Matrices provide an alternative to ordinary linear algebra which allows us to deal with multi-dimensional data in a more concise way.
  • They are similar in form to a table, with a number of entries arranged into rows and columns.
  • The size of a matrix is expressed in the form rows x columns (i.e. the number of rows followed by the number of columns, with a cross separating them).
  • We can refer to a specific element in a matrix using the name of the matrix, with a subscript listing the row and column corresponding to the element in question e.g. for a matrix; A, the element in the 2nd row and 1st column is denoted by A_{2,1}
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