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}.

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 2^{nd} row and 1^{st} column is denoted by A_{2,1}