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
