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 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: