23 Applications of Determinants

Sachin Kumar

Objectives:

1. To familiarize the students with Cramer’s rule for solving system of linear equations

2. To find the area of triangle using determinants

3. To derive the condition for three points to be collinear

4. To derive the equation of a line passing through given points

Introduction to Applications of Determinants

Determinants have many applications in various fields. Determinants are used to check singular and non-singular matrices and if non-singular then it is used to find the inverse of matrix. Using determinants, linear system of equations can be solved. Using determinants when can examine weather given system of linear equations have unique solution. So determinants have various applications. Determinants also play important role in the field of environmental sciences. Some of important applications of determinants will be discussed in this module.

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If a system has one or more solutions then it is said to be consistent system of equations.

If system has no solution then it is said to be inconsistent system of equations.

Methods for solving system ?? = ?

First we will discuss to solve a non-homogeneous system of simultaneous linear equations by Cramer’s rule named after the Swiss mathematician Gabriel Cramer

Cramer’s Rule for Non-homogeneous System:

The solution of the system of linear equations

Step (iv) Take any two equations out of given three equation from system and shift one of the variable, say ?, on the right hand side two obtain two equations in ?, ?. Solve these two equations to obtain ?, ? in terms of ?.

If these values of ?, ? satisfies the third equation of system, then the system is consistent and have infinite many solutions and values of x, y and z in terms of arbitrary constant ? constitute a solution.

If values of ?, ? in terms of ? does not satisfy third equation of system then system is inconsistent and have no solution.

Example 1 (Unique solution). Solve by Cramer’s rule:

So obtain the solution of this system by Cramer’s rule in terms of ?.

The solution obtained with ? = ? gives a solution of given system. So for each value of ? , we have solution of given homogeneous system. Hence we obtain infinitely many solutions.

Example 1. Solve the following system of homogeneous equations

3? − 4? + 5? = 0,

? + ? − 2? = 0,

2? + 3? + ? = 0.

Solution:

Example: For which value of ?, the homogeneous system of linear equations:

4? − 2? + 3? = 0

? + ?? + 2? = 0

2? + 2? = 0

has non-trivial solutions and also find the corresponding solution of system.

Solution: The given homogeneous system will have non-trivial solution if

⇒ 4(2? − 0) + 2(2 − 4) + 3(0 − 2?) = 0

⇒ 2? − 4 = 0

⇒ ? = 2.

So given system of equations will have non-trivial solutions for ? = 2.

Now take first two equations of the system and put ? = ?, where ? is arbitrary constant, we have

4? − 2? = −3?

? + 2? = −2?

Again we will solve this system using Cramer’s rule.

So last equation is also satisfied.

So non-trivial solution of the given system is

Summery:

In this module, we have shown some applications of determinants. We have deduced the criteria of finding the area of triangle using determinants. Then we have also deduced the condition for three points in a plane to be collinear in the form of determinant. It is also shown that using determinants, equation of line passing through given points can be determined easily. Then Cramer’s rule for solving system of linear equations is introduced. Various examples have also been discussed to understand the applications 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.