22 Determinants and its Properties

Sachin Kumar

Objectives:

1. To familiarise the students with Determinants, Singular and Non-Singular Matrices

2. To let the students understand some properties of determinants

3. To let the students evaluate determinants using properties.

    Introduction to Determinants

Determinants are very important as they have many applications. Determinants are used to check the nature of matrix weather it is invertible or not and also if invertible then it is used for finding the inverse of matrix. Using determinants in Cramer’s rule, we can solve linear system of equations. It also helps to check the nature of solutions of system of linear equations. Determinants are also used to find area of triangle and volume of tetrahedron and also for finding the equation of lines and planes. So determinants are very useful.

As matrices have many application in environment sciences, so determinants are also important tool in environment sciences. Most of the characteristics of matrices are determined by determinant of matrices. We will discuss in this module that when will a matrix be singular or non-singular. Only non-singular matrices have multiplicative inverse. So determinants play important role for applications of matrices in environmental sciences.

DEFINITION: Every square matrix can be associated to an expression or a number which is known as its determinant. If ?=[???] is a square matrix of order ?, then the determinant of ? is denoted by det??? |?| ??,

Thus the determinant of a square matrix of order 3 is the sum of the product of elements of ?_1? in first row with (−1)^1+? times the determinant of a 2×2 sub matrix obtained by leaving the first row and column passing through the element ?1?.

Determinant of any square matrix of order ?

Let ? be any square matrix of order ?≥2, then determinant of ? is defined as

Theorem: If is an × matrix, then the determinant of can be computed by multiplying the entries of any row (or column) by their cofactors and summing the resulting products:

obtained from matrix ? by multiplying second row of matrix ? by ? and then adding to first row. It is easy to check that |?|=40.

So for simplifying the determinant expansion, using the properties of determinants, we will perform following row/column operations:

Interchanging the rows

?th row with in matrix can be changed with ?th row of matrix denoted by ?_?↔?_?.

Multiplying a row with scalar

Elements in a row can be multiplied by non-zero scalar.

?_?→??_?, where ?≠?.

Addition of rows

A row ?_? in a matrix can be changed with sum of row ?_? with scalar multiple of another row ?_? i.e.

?_?+??_?→?_?

Similarly same operations can be performed for columns as well

Some illustration of evaluation of determinants using properties of determinants

Example. Evaluate without expanding the following determinant:

Summary:

In this module, we have introduced the concept of determinants of any order square matrix. We have also introduced the concept of minors and cofactors of square matrix. Then we have also discussed some properties of determinants. Using these properties, determinant can be evaluated easily. Various examples have also been discussed to understand the properties of determinants.

Suggested Books for reading:

[1] Kreyszig, Erwin. Advanced engineering mathematics. John Wiley & Sons, 2010.

[2] Leon, Steven J. Linear algebra with applications. Pearson Prentice Hall, 2006.

[3] Bronson, Richard. Matrix Methods: An Introduction. Academic Press, 1970

[4] Lay, David C. Linear algebra and its applications, Addison Wesley, 2005