Matrices And Determinants: Definition, Formulas, Examples

In linear algebra, matrices and determinants are used to solve linear equations by applying Cramer’s rule to a collection of non-homogeneous linear equations. This chapter introduces students to the tool known as Matrix or Matrices, which help solve multiple equations simultaneously. Furthermore, the chapter Determinants introduces students to how Determinants are used to represent functions associated with Matrices. Since both chapters, Matrices and Determinants (Chapters 3 and 4 in Class 12 Textbook), are very interlinked, they are often referred to as Matrices and Determinants collectively.

In this article, students will be able to understand why the chapter is important, know important sub-topics to study, learn a few basic properties of Matrices and Determinants, best books to study the topics, and links to practice for the chapter online. Read the article to know more:

Matrices And Determinants Definitions

Before getting into all the details and resources about Matrices and Determinants, let’s understand briefly what they are:

What Are Matrices?

A set of numbers objects or symbols represented in the form of a rectangular array is called a matrix. The order of the matrix is defined by the number of rows and number of columns present in the rectangular array of representation. As displayed in the following image, there are two different matrices with the same numbers but in a different structure.

What Are Determinants of Matrices?

To every square matrix A = [aij] of order n, we can associate a number (real or complex) called the determinant of the square matrix A, where aij = (i, j)th element of A. It is crucial to understand Matrices in order to understand determinants and how they are used to perform calculations with Matrices. Furthermore, knowledge of the determinants is very useful for almost all the topics of Mathematics, especially in vectors, coordinate geometry and calculus.

Important Topics in Matrices And Determinants

Since the concepts of Matrices and Determinants are useful in later chapters related to algebra, it is important to thoroughly cover the chapter for all students. However, there are some sub-topics you should definitely not skip while learning the chapter.

Matrix and operation on matrices
Types of matrix
Transpose of a matrix, symmetric and skew-symmetric matrix
Conjugate of matrix, hermitian and skew-hermitian matrix
Determinant of matrix
Minor and cofactor of an element of matrix/determinant
Adjoint and inverse of a matrix
Elementary row operations and its use in finding the inverse of a matrix
System of linear equations and Cramer’s rule
System of homogeneous linear equations

Best Books To Study

After you learn the basics of Matrices and Determinants, you can refer some books that offer you advanced knowledge on the topic and help you master it. Here are some of the best books to study Matrices and Determinants:

Algebra by S.K. Goyal [Arihant]

TMH’s Comprehensive Mathematics for JEE Advanced

Higher Algebra by Hall & Knight [Arihant]

Important Formulas

Any matrix A and its transpose both have the same Eigen values.
The trace of the matrix equals the sum of the Eigen values of a matrix.
The determinant of the matrix A equals to the product of the Eigen values of A.
If λ₁, λ₂, ….., λ_n are the n-Eigen values of A, then

Cayley Hamilton Theorem

Let A be an n x n matrix and let

Matrix exponential for a square matrix A,

Practice Questions

To score good marks in JEE Mathematics, it is important for students to cover this chapter during preparation for IIT JEE. Accordingly, students need to practice constantly so they can cover all the different kind of problems and questions related to Matrices and Determinants. Emb i be‘s AI-enabled system is richly populated with questions from all topics for the JEE exam. Accordingly, students can attempt different chapter tests for Matrices and Determinants, and can also take complete JEE Main Mock Tests when students have covered the complete syllabus.

Probability

Set Theory And Relations

Addition and Subtraction of a Matrix

Adjoint of A Matrix

Classification Of Matrices

Column Matrix

FAQs

Question: What is meant by determinant?
Answer: The determinant is defined as a scalar value which is associated with the square matrix. If X is a matrix, then the determinant of a matrix is represented by |X| or det (X).

Question: Why do we use determinants?
Answer: The determinants are used to solve the system of linear equations and it is also used to find the inverse of a matrix.

Question: What is a matrix?
Answer: A matrix is defined as the rectangular array of numbers. The collection of numbers are arranged in rows and columns.

Question: Are matrices important for JEE?
Answer: Matrices are the most of the easiest chapter of entrance examinations, but at the same time they plays an important role in JEE Main and JEE Advanced.

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