This module covers the definition of matrices, different types of matrices, matrix operations, transition matrices and the use of first-order linear matrix recurrence relations to model a range of situations and solve related problems.

Matrices and their applications, including:

- review of matrix arithmetic: the order of a matrix, types of matrices (row, column, square, diagonal, symmetric, triangular, zero, binary and identity), the transpose of a matrix, elementary matrix operations (sum, difference, multiplication of a scalar, product and power)
- inverse of a matrix, its determinant, and the condition for a matrix to have an inverse
- use of matrices to represent numerical information presented in tabular form, and the use of a rule for the \( {a_{ij}}^{th} \) element of a matrix to construct the matrix
- binary and permutation matrices, and their properties and applications
- communication and dominance matrices and their use in analysing communication systems and ranking players in round-robin tournaments
- use of matrices to represent systems of linear equations and the solution of these equations as an application of the inverse matrix; the concepts of dependent systems of equations and inconsistent systems of equations in the context of solving pairs of simultaneous equations in two variables; the formulation of practical problems in terms of a system of linear equations and their solution using the matrix inverse method.

Transition matrices, including:

- use of the matrix recurrence relation: \( S_0 = \) initial state matrix, \( S_{n+1} = TS_n \) where \( T \) is a transition matrix and \( S_n \) is a column state matrix, to generate a sequence of state matrices, including in the case of regular transition matrices an informal identification of the equilibrium state matrix (recognised by no noticeable change from one state matrix to the next)
- use of transition diagrams, their associated transition matrices and state matrices to model the transitions between states in discrete dynamical situations and their application to model and analyse practical situations such as the modelling and analysis of an insect population comprising eggs, juveniles and adults
- use of the matrix recurrence relation \( S_0 = \) initial state matrix, \( S_{n+1} = TS_n + B \) to extend the modelling to populations that include culling and restocking.

*source – VCE Mathematics Study Design*

Lessons

#### Properties of Matrix Multiplication

- Topic - Commutative Law of Matrices
- Topic - Commutative Law with Zero Matrix
- Topic - Verifying Distributivity of Matrix Multiplication using 2 by 2 Matrices
- Topic - Associative Law of Matrix Multiplication using 2 by 2 Matrices
- Topic - Identify Matrix of Multiplication using 2 by 2 Matrices
- Topic - Square of 2 by 2 Matrices
- Topic - Cube of 2 by 2 Matrices
- Topic - Square of 3 by 3 Matrices
- Topic - Conditions of Square a Matrix