Determinants
UNIT 1 DETERMINANTS
Structure
1.0 Introduction
1.1 Objectives
1.2 Determinants of Order 2 and 3
1.3 Determinants of Order 3
1.4 Properties of Determinants
1.5 Application of Determinants
1.6 Answers to Check Your Progress
1.7 Summary
1.0 INTRODUCTION
In this unit, we shall learn about determinants. Determinant is a square array of
numbers symbolizing the sum of certain products of these numbers. Many
complicated expressions can be easily handled, if they are expressed as
‘determinants’. A determinant of order n has n rows and n columns. In this unit,
we shall study determinants of order 2 and 3 only. We shall also study many
properties of determinants which help in evaluation of determinants.
Determinants usually arise in connection with linear equations. For example, if
the equations a1x + b1 = 0, and a2x + b2 = 0 are satisfied by the same value of x,
then a1b2 – a2 b1 = 0. The expression a1b2 a2b1 is a called determinant of
second order, and is denoted by
There are many application of determinants. For example, we may use
determinants to solve a system of linear equations by a method known as
Cramer’s rule that we shall discuss in coordinate geometry. For example, in
finding are of triangle whose three vertices are given.
1.1 OBJECTIVES
After studying this unit, you should be able to :
define the term determinant;
evaluate determinants of order 2 and 3;
use the properties of determinants for evaluation of determinants;
use determinants to find area of a triangle;
use determinants to solve a system of linear equations (Cramer’s Rule)
5
,Algebra - I
1.2 DETERMINANTS OF ORDER 2 AND 3
We begin by defining the value of determinant of order 2.
Definition : A determinant of order 2 is written as where a,b, c, d are
complex numbers. It denotes the complex number ad – bc. In other words,
Example 1 : Compute the following determinants :
(a) (b)
(c) (d)
Solutions :
(a) = 18 − (–10) = 28
(b) = – =0
(c) = +
( (a + ib) (a − ib) = )
6
, Determinants
1.3 DETERMINANTS OF ORDER 3
Consider the system of Linear Equations :
x+ y+ z= ……………….. (1)
x+ y+ z= ..……………….. (2)
x+ y+ z= ……………….. (3)
Where aij C ( 1 ≤ i, j ≤ 3) and , , , C Eliminating x and y from these
equation we obtain
We can get the value of z if the expression
– – –
The expression on the L.H.S. is denoted by
and is called a determinant of order 3, it has 3 rows, 3 columns and is a complex
number.
Definition : A determinant of order 3 is written as
where aij C (1 ≤ i, j ≤ 3).
It denotes the complex number
+
Note that we can write
= +
= – –
= – +
7
UNIT 1 DETERMINANTS
Structure
1.0 Introduction
1.1 Objectives
1.2 Determinants of Order 2 and 3
1.3 Determinants of Order 3
1.4 Properties of Determinants
1.5 Application of Determinants
1.6 Answers to Check Your Progress
1.7 Summary
1.0 INTRODUCTION
In this unit, we shall learn about determinants. Determinant is a square array of
numbers symbolizing the sum of certain products of these numbers. Many
complicated expressions can be easily handled, if they are expressed as
‘determinants’. A determinant of order n has n rows and n columns. In this unit,
we shall study determinants of order 2 and 3 only. We shall also study many
properties of determinants which help in evaluation of determinants.
Determinants usually arise in connection with linear equations. For example, if
the equations a1x + b1 = 0, and a2x + b2 = 0 are satisfied by the same value of x,
then a1b2 – a2 b1 = 0. The expression a1b2 a2b1 is a called determinant of
second order, and is denoted by
There are many application of determinants. For example, we may use
determinants to solve a system of linear equations by a method known as
Cramer’s rule that we shall discuss in coordinate geometry. For example, in
finding are of triangle whose three vertices are given.
1.1 OBJECTIVES
After studying this unit, you should be able to :
define the term determinant;
evaluate determinants of order 2 and 3;
use the properties of determinants for evaluation of determinants;
use determinants to find area of a triangle;
use determinants to solve a system of linear equations (Cramer’s Rule)
5
,Algebra - I
1.2 DETERMINANTS OF ORDER 2 AND 3
We begin by defining the value of determinant of order 2.
Definition : A determinant of order 2 is written as where a,b, c, d are
complex numbers. It denotes the complex number ad – bc. In other words,
Example 1 : Compute the following determinants :
(a) (b)
(c) (d)
Solutions :
(a) = 18 − (–10) = 28
(b) = – =0
(c) = +
( (a + ib) (a − ib) = )
6
, Determinants
1.3 DETERMINANTS OF ORDER 3
Consider the system of Linear Equations :
x+ y+ z= ……………….. (1)
x+ y+ z= ..……………….. (2)
x+ y+ z= ……………….. (3)
Where aij C ( 1 ≤ i, j ≤ 3) and , , , C Eliminating x and y from these
equation we obtain
We can get the value of z if the expression
– – –
The expression on the L.H.S. is denoted by
and is called a determinant of order 3, it has 3 rows, 3 columns and is a complex
number.
Definition : A determinant of order 3 is written as
where aij C (1 ≤ i, j ≤ 3).
It denotes the complex number
+
Note that we can write
= +
= – –
= – +
7
