• UNIT 12 : Matrices and determinants of order 2

    Key unit competence 

    Use matrices and determinants of order 2 to solve systems of linear equations and to define transformations of 2D.

    Learning objectives 



    12.1 Introduction

    Activity 12.1 

    In pairs, carry out research to find out the meaning of the term matrix. What is the plural of matrix? Where and when can we make use of a matrix?

    A matrix is an ordered set of numbers listed in rectangular form. 

    Order of a matrix 

    The number of rows and columns that a matrix has is called its order or its dimension. By convention, rows are listed first; and columns, second. Thus, we would say that the order (or dimension) of the matrix below is 3 x 4, meaning that it has 3 rows and 4 columns. 



    Square matrix 


    Diagonal matrix 

    A diagonal matrix is a square matrix with all non-diagonal elements 0. The diagonal matrix is completely defined by the diagonal elements.


    Row matrix 

    A matrix with one row is called a row matrix. 

    [2   5    –1   5]


    Column matrix 

    A matrix with one column is called a column matrix. 


    We talk about one matrix, or several matrices. There are many things we can do with them.


    12.2 Matrix of a linear transformation 



    12.3 Matrix of geometric transformation 

    Symmetric about x-axis and y-axis 



    Rotation




    12.4 Operations on matrices 



    Equality of matrices

    Activity 12.3 

    Carry out research to find out the meaning of equal matrices. Discuss in class using examples.

    For two matrices to be equal, they must be of the same size and have all the same entries in the same places. For instance, suppose you have the following two matrices:


    These matrices cannot be the same, since they are not of the same size. If A and B are the following two matrices:




    Addition and subtraction of matrices 

    Addition and subtraction of matrices is only possible for matrices of the same order.

    Addition 

    To add two matrices: add the numbers in the matching positions:


    The two matrices must be of the same size, i.e. the rows must match in size, and the columns must match in size.



    Subtraction 

    Negative 

    The negative of a matrix is also simple:




    Multiplication of a matrix by a scalar 

    To multiply a matrix by a single number is easy:




    Identity matrix 


    Multiplication of matrices

    Two matrices can be multiplied if and  only if the number of columns of the first matrix is equal to the number of rows of the second matrix. Such matrices are said to be compatible to multiplication. 

    To multiply a matrix by another matrix we need to use the dot product of rows and columns. 

    Let us see using an example: 




    Transpose of a matrix 



    12.5 Determinant of a matrix of order 2

    Activity 12.4

    In pairs, find out the meaning of the term determinant as used in the case of matrices. Why do we need determinants? What is the symbol for determinant?

    The determinant of a matrix is a special number that can be calculated from a square matrix. The symbol for determinant is two vertical lines either side. For example, 



    Inverse of matrices 

    If A is a square matrix then a matrix B such that AB = BA = I2 is called an inverse of the matrix A. 





    Application of determinants 

    Cramer’s method 









    UNIT 11 : Concepts and operations on linear transformations in 2DUNIT 13 : Points, straight lines and circles in 2D