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NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.4 Determinants - PDF Download

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NCERT solutions class 12 maths chapter 4 exercise 4.4 Determinants describes the essential concepts of adjoint and inverse of a matrix. You will get to learn four theorems in this NCERT solution which are based on concepts of adjoint of a matrix. Students will find it easier to apply these theorems to get the inverse of a matrix if they refer to the NCERT solutions in class 12 maths chapter 4 ex 4.4. 

There are a total of 18 questions in class 12 maths chapter 4 exercise 4.4 NCERT solutions that are based on finding the inverse of matrix using adjoint. Ex 4.4 class 12 maths chapter 4 solutions are prepared by the academic team of mathematics at eSaral that will help you to solve questions of ex 4.4 in stepwise manner. Students can gain an in-depth knowledge of the methods and theorems to find the adjoint and inverse of determinants by going through the NCERT solutions. 

Ex 4.4 class 12 maths chapter 4 is also available in PDF format that can be downloaded from eSaral website. You can download the solution PDFs for free from the given link below and practice all the questions of ex 4.4 to achieve a high score in board exams.

Topics Covered in Exercise 4.4 Class 12 Mathematics Questions

Ex 4.4 class 12 maths chapter 4 Determinants is all about finding the adjoint and inverse of a matrix. Detailed explanation is provided below

1.

Adjoint and Inverse of a Matrix

  • Adjoint of a matrix and Theorems

  1. Adjoint and Inverse of a Matrix 

In the last chapter, we have studied the inverse of a matrix. Here, we shall discuss the condition for existence of inverse of a matrix. 

We must first define the adjoint of a matrix in order to get the inverse of a matrix A, i.e. A–1.

  • Adjoint of a matrix 

Definition 1- The adjoint of a square matrix A = [aij]n × n is defined as the transpose of the matrix [Aij]n × n , where Aij is the cofactor of the element aij. Adjoint of the matrix A is denoted by adj A.

Theorem 1:  If A be any given square matrix of order n, then

                     A(adj A) = (adj A) A = A I, where I is the identity matrix of order n

Definition 2- A square matrix A is said to be singular if A = 0.

For example, the determinant of matrix A = $\left[\begin{array}{ll}1 & 2 \\ 4 & 8\end{array}\right]$  is zero, hence A is a singular matrix. 

Definition 3- A square matrix A is said to be non-singular if A ≠ 0 

Suppose $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$ then $|A|=\left|\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right|=4-6=-2 \neq 0$.

Hence A is a nonsingular matrix.

Here are the following theorems without proof based on the concepts mentioned above.

Theorem 2: If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.

Theorem 3: The determinant of the product of matrices is equal to product of their respective determinants, that is, AB = A B , where A and B are square matrices of the same order

 if A is a square matrix of order n, then |adj(A)| = |A|n – 1 .

Theorem 4: A square matrix A is invertible if and only if A is nonsingular matrix

Tips for Solving Exercise 4.4 Class 12 Chapter 4 Determinants

NCERT solutions for ex 4.4 class 12 maths chapter 4 has questions based on the topics of adjoint and inverse of determinants. To solve these questions our expert teachers of eSaral have provided some tips which are framed in simple and easy to comprehend language.  

  1. students should attempt every question of ex 4.4 in NCERT solutions to ensure they fully grasp the concept. This will help them to develop an understanding to get an idea of types of questions that can be asked during exams.

  2. You should not only learn the concepts of adjoint and inverse of matrices but also learn all the theorems provided in these solutions to solve questions without any doubt.

  3. It is recommended to you to practice the examples provided in NCERT solutions to get a deep knowledge of concepts.

Importance of Solving Ex 4.4 Class 12 Maths Chapter 4 Determinants

There are numerous benefits of solving ex 4.4 class 12 maths chapter 4 determinants. Some of the benefits are explained by our experts of eSaral that you can check below.

  1. NCERT solutions for ex 4.4 class 12 maths ch 4 has explained each topic in an easy to comprehend manner so that you can solve all questions with proper understanding.

  2. Class 12 maths chapter 4 ex 4.4 NCERT solutions questions can be solved to help you determine how much time you should spend on each topic.

  3. By solving ex 4.4 class 12 maths chapter 4, you can improve your problem solving skills.

  4. Practicing questions of ex 4.4 frequently and revising concepts, you will be able to gain conceptual clarity of topics that will help you score good marks in examinations.

  5. Solving exercise 4.4 class 12 maths chapter 4 NCERT solutions will also boost your confidence to write solutions in board exams

Frequently Asked Questions 

Question 1. Is a non-singular matrix invertible?

Answer 1. Yes, a non-singular matrix is invertible.

Question 2. What is a singular matrix?

Answer 2.  A square matrix A is said to be singular if A = 0.

For example, the determinant of matrix $A=\left[\begin{array}{ll}1 & 2 \\ 4 & 8\end{array}\right]$  is zero. Hence A is a singular matrix.

 

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