Properties of Determinants

What is a Determinant?

Determinant 

Every square matrix has an association with a scalar which is called a determinant. 

A determinant is denoted by writing det(A) or I A I.

Note: The determinant value of a matrix with order 1 is its scalar value. But, the determinant of a matrix with order 2 x 2 is the difference of its cross products. 

A Determinant of a given matrix shows only a single number. A student can find the value of a determinant by adding and multiplying its elements in a specially organized way.

Determinants are regarded as a matrix scaling factor. They may be regarded as matrices that try to expand and decrease in terms of functions. Determinants use a square matrix as input and give us an output that is just a single number.

Properties of Determinants

There are various Properties/ Attributes related to the solution of Determinants.

Property 1: The solution of a given determinant remains the same if its columns and rows are interchanged.

Property 2: If any of the two columns or rows of a given determinant are interchanged, then the sign of the given determinant is also changed.

For example- In case, 2 rows in a determinant are interchanged then the value of the determinant becomes negative (-ve).

Property 3: If any of the two columns or rows of a given determinant are the same that means that every element corresponding to the other one is similar, then the value of the determinant is considered zero.

Property 4: If all the elements present in a column or row of a given determinant are multiplied by a constant, for example, ‘k’, then the value of the determinant is also multiplied by ‘k’.

Property 5: If few or all of the elements present in a column or row of a determinant are written in the form of the sum of two or more terms, then the determinant can also be written as the sum of two or more determinants.

Property 6: If all the elements are present in any column or row of a given determinant, then the multiples of the respective elements of another column or row are added, the value of the determinant still remains the same.

Property 7: If all the elements present in a specific column or row are zeros then the determinant value is 0.

Property 8: The value of the determinant of a triangular matrix (lower triangular matrix or upper triangular matrix) is equal to the value of the multiplication elements that are present in the diagonal.

Property 9: If the value of the order of X and Y is the same, then

Det (XY) = Det (X) Det (B)

Calculations of Determinants

If we have a simple square matrix of order 1×1, that has only one number, then the value of the determinant becomes the given number itself. 

In the case of any square matrix of an order 2×2, a student can use the formula of determinants to calculate the value of determinants.

For the above-given matrix let us consider that

p = 1

q = 2

r = 3

s = 4

Then the determinant can be found by finding the subtraction of the cross products of the elements in the given matrix.

Solution

IAI = (p x s) – (r x q)

      = (1 x 4) – (3 x 2)

      = (4) – (6)

      = 4 – 6

      = -2.

Conclusion

Determinants and their properties provide us with a lot of help when solving problems related to Determinants, it also shows us an extremely linear and easy path for solving hectic equations in day to day life, especially in the field of algebra.

The laws and solutions of determinants are used for plotting graphs, statistical studies and also to do scientific studies and research in almost different fields. Determinants can also be used to generate and show us data of the world in real life like the population of people, infant mortality rate, population per square meter, deaths per year, etc. They are the best representation methods for plotting and performing surveys.

This module will surely help and make it easy for the students to perform solutions and boost their morale for mathematical examinations, not only at school levels but also in competitive examinations.