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2.1 Simple Matrix Recurrence Relations

The Simple Matrix Recurrence Relation Formula

  • The simplest type of matrix recurrence relation formula we will analyse in Further Maths models a system where the next “state”; S_{n+1}, can be reached by multiplying the current state; S_{n}, by a transition matrix; T, in the form:

S_{n+1}=T S_{n}

  • The state matrices; S_{n} (where n is a positive whole number representing the state of the system), are column matrix listing the value of each of the system’s variables in the corresponding state.
  • The transition matrix; T, is a square matrix.
  • As with a linear recurrence relation, it is important to state the initial state of a system; S_{0}.
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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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1.1 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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