Are you looking for Chapter 4 Determinants Class 12 NCERT Solutions 4 Free PDF? So today you are going to get all the things under this article. Today I am going to give you the solutions of determinants chapter of class 12th.
The solution that I am going to give you is from NCERT Solutions for Class 12 Maths Chapter 4, inside which the solution of all the exercises will be present. Today I will also give you detailed information about this chapter.
- Determinants Exercise 4.1
- Determinants Exercise 4.2
- Determinants Exercise 4.3
- Determinants Exercise 4.4
- Determinants Miscellaneous Exercise
Determinants are an important topic in Mathematics, and Chapter 4 of Class 12 NCERT Solutions provides a comprehensive understanding of this subject. In this chapter, students will learn how to evaluate determinants, find their properties and applications, and solve problems related to matrices and determinants.
The chapter starts with an introduction to the concept of determinants, explaining what they are and why they are important. Students will learn that determinants are a special type of function that takes in a matrix as an input and produces a scalar value as an output. This scalar value is a measure of the matrix’s properties, such as its invertibility, linear dependence, or area/volume.
The next section of the chapter covers the evaluation of determinants. Here, students will learn the different methods of calculating determinants, such as expansion by minors, cofactor expansion, and using row or column operations. The chapter explains each method in detail and provides examples to help students understand the concepts better.
You Can Also Read: Maths NCERT Solutions For Class 12 Chapter 3 Matrices
After evaluating determinants, the chapter moves on to discuss the properties of determinants. Here, students will learn about various properties such as linearity, scalar multiplication, and Transpose of a matrix. The chapter explains each of these properties in detail and provides examples to help students understand how they work.
The chapter then moves on to the topic of the adjoint and inverse of a matrix. Students will learn the relationship between the determinant of a matrix and its inverse, as well as how to find the inverse of a matrix using its adjoint. The chapter also covers the uniqueness of the inverse of a matrix and the conditions for the existence of an inverse.
Finally, the chapter concludes with some applications of determinants. Students will learn how to use determinants to solve problems related to systems of linear equations, area and volume, and the inverse of a matrix. The chapter provides examples and exercises to help students understand how to apply determinants to solve problems.
Determinants Class 12 NCERT Solutions 4.1 Exercise
Determinants Class 12 NCERT Solutions 4.1 is a set of questions that test the student’s understanding of the evaluation and properties of determinants. The exercise consists of 15 questions, with multiple parts, that require students to apply different methods to find the value of determinants.
The questions cover various topics such as expansion by minors, cofactor expansion, the properties of determinants, adjoint, and inverse of matrices. By attempting these questions, students can reinforce their knowledge of determinants and sharpen their problem-solving skills.
What are Determinants?
Determinants are a mathematical concept that is used to measure the properties of matrices. They are a special type of function that takes in a square matrix as input and produces a scalar value as output. This scalar value is a measure of the matrix’s properties, such as its invertibility, linear dependence, or area/volume.
The determinant of a matrix is denoted by vertical bars on either side of the matrix, such as |A|. Determinants play an important role in many fields of mathematics, including linear algebra, differential equations, and calculus. They have numerous applications in physics, engineering, and computer science, such as solving systems of linear equations, finding the area of a triangle in 2D or the volume of a parallelepiped in 3D, and computing the inverse of a matrix.
Difference Between Determinants and Matrix
Here is a table summarizing the main differences between a matrix and a determinant:
| Matrix | Determinant | |
|---|---|---|
| Definition | A rectangular array of numbers arranged in rows and columns | The scalar value computed from a square matrix |
| Notation | Denoted by a capital letter (e.g., A, B, C) | Denoted by Det(A) or |
| Size | Can have any number of rows and columns | Must be a square matrix (same number of rows and columns) |
| Purpose | Used to represent and manipulate linear equations, transformations, and other mathematical concepts | Used to determine properties of matrices, such as invertibility |
| Calculation | Elements of the matrix are used to perform various operations, such as addition, subtraction, and multiplication | A formula involving the elements of the matrix is used to compute the determinant |
| Result | A matrix can be transformed through operations to obtain a new matrix with different properties | The determinant is a scalar value that encodes important information about the matrix, such as whether the matrix is invertible or not |
In summary, matrices and determinants are both important mathematical concepts, but they have different properties and purposes. Matrices are used to represent and manipulate linear equations, while determinants are used to determine properties of matrices, such as invertibility.
Determinants Class 12 NCERT Solutions 4.2 Exercise Chapter 4
Determinants Class 12 NCERT Solutions 4.2 is a crucial part of the mathematics syllabus that deals with determinants. This exercise is designed to test students’ knowledge and understanding of evaluating determinants using various methods. It comprises 12 questions with multiple parts that require students to apply different techniques to determine the value of determinants.
In this exercise, students will encounter questions that cover the cofactor expansion method, properties of determinants, and the adjoint and inverse of matrices. By attempting these questions, students can sharpen their problem-solving skills and reinforce their understanding of determinants.
One of the essential topics covered in Exercise 4.2 is the calculation of determinants using the cofactor expansion method. This method involves expanding the determinant along a row or column and computing the cofactors of each element. This technique is particularly useful for finding the value of determinants of higher-order matrices.
Another important topic in this exercise is the properties of determinants. Students will learn about the determinant of the transpose of a matrix, the product of matrices, and the determinant of the inverse of a matrix. These properties help students to solve complex problems and understand the underlying concepts better.
Properties of Determinants for Class 12th
Properties of determinants are essential concepts in mathematics, particularly in the study of matrices. Class 12 students are introduced to these properties as they learn about the evaluation and manipulation of determinants. Here are some of the important properties of determinants for Class 12:
- The determinant of a matrix remains the same when its rows and columns are interchanged.
- If two rows (or columns) of a matrix are interchanged, the determinant changes sign.
- If any two rows (or columns) of a matrix are proportional, the determinant becomes zero.
- If each element of a row (or column) of a matrix is multiplied by a scalar k, the determinant is multiplied by k.
- If two rows (or columns) of a matrix are identical, the determinant is zero.
- If a matrix has a row (or column) consisting entirely of zeros, its determinant is zero.
- The determinant of the product of two matrices is equal to the product of their determinants.
- The determinant of the Transpose of a matrix is equal to the determinant of the original matrix.
- The determinant of the inverse of a matrix A is the reciprocal of the determinant of A, provided A is invertible.
- The determinant of a diagonal matrix is equal to the product of its diagonal entries.
- The determinant of an upper triangular matrix is equal to the product of its diagonal entries.
- The determinant of a lower triangular matrix is equal to the product of its diagonal entries.
By understanding and applying these properties, students can efficiently evaluate the determinants of matrices and solve complex problems in linear algebra and other related fields.
Determinants Class 12 NCERT Solutions 4.3 Exercise Chapter 4
Determinants Class 12 NCERT Solutions 4.3 Exercises focus on the concept of determinants and their applications in solving systems of linear equations. In this exercise, students will learn about the inverse of a matrix, Cramer’s rule, and the use of determinants in solving problems involving matrices.
Determinants play a crucial role in linear algebra as they can be used to determine the invertibility of a matrix and help in solving systems of linear equations. Exercise 4.3 provides students with the opportunity to practice their problem-solving skills and enhance their understanding of these concepts.
In this exercise, students will also learn about Cramer’s rule, which is an efficient method for solving systems of linear equations using determinants. By applying this rule, students can find the unique solution of a system of equations, given that the determinant of the coefficient matrix is non-zero.
Through Exercise 4.3, students will also learn about the inverse of a matrix, which is a fundamental concept in linear algebra. They will understand how to calculate the inverse of a matrix using the adjoint method and apply this knowledge in solving complex problems.
Minor and Co – factor
In Determinants Class 12 NCERT Solutions 4.3, students learn about the concept of determinants and their applications in linear algebra. One of the essential concepts in the study of determinants is minors and cofactors.
A minor of a matrix is the determinant of a submatrix obtained by deleting some rows and columns of the original matrix. For example, the 2×2 matrix obtained by deleting the first row and first column of a 3×3 matrix is a minor of that matrix.
A cofactor is the signed minor of a matrix element. The sign of the cofactor depends on the position of the element in the matrix. If the element is in an odd position, the cofactor is negative, and if it is in an even position, the cofactor is positive.
Using minors and cofactors, we can calculate the determinant of a matrix using the Laplace expansion method. This method involves expanding the determinant of a matrix in terms of its minors and cofactors along a particular row or column.
Moreover, minors and cofactors are useful in finding the inverse of a matrix. To find the inverse of a matrix, we first find its adjoint matrix, which is obtained by taking the transpose of the matrix of cofactors. Then, we divide the adjoint matrix by the determinant of the original matrix to get the inverse.
Chapter 4 Determinants Class 12 NCERT Solutions 4.4 Exercise
Class 12 NCERT Solutions Chapter 4 Exercise 4.4 focuses on the properties of determinants and their applications in linear algebra. This exercise deals with various properties of determinants, such as the addition and subtraction of rows and columns, and their applications in solving systems of linear equations.
The exercise begins by introducing the concept of row and column operations, which involve adding or subtracting one row or column from another. These operations do not change the value of the determinant and can be used to simplify the calculation of determinants.
In this exercise, students will learn about the properties of determinants that enable us to calculate the determinant of a matrix quickly. One of the crucial properties is the multiplication of a row or column by a constant, which multiplies the determinant by the same constant.
The exercise also covers the concept of singular and non-singular matrices. A matrix is non-singular if its determinant is non-zero, and it is singular if its determinant is zero. Non-singular matrices have an inverse, which can be used to solve systems of linear equations.
The application of determinants in solving systems of linear equations is an essential part of this exercise. Students will learn how to use Cramer’s rule, which involves the calculation of determinants, to solve a system of linear equations. They will also learn how to use the inverse of a matrix to solve systems of linear equations efficiently.
Adjoint and Inverse of a Matrix class 12
Determinants Class 12 NCERT Solutions 4.4, students learn about the concept of determinants and their applications in linear algebra, including the adjoint and inverse of a matrix.
The adjoint of a matrix is obtained by taking the transpose of the matrix of cofactors of a given matrix. The matrix of cofactors is obtained by taking the determinant of a submatrix obtained by deleting the row and column containing a particular element of the original matrix and multiplying it by -1 if the sum of the row and column indices is odd.
The adjoint matrix is used to find the inverse of a non-singular matrix, which is a matrix that has a non-zero determinant. The inverse of a non-singular matrix is defined as the product of the adjoint matrix and the reciprocal of the determinant of the original matrix.
The inverse of a matrix is essential in many applications of linear algebra, such as solving systems of linear equations and finding the eigenvalues and eigenvectors of a matrix. For example, in solving a system of linear equations, we can use the inverse of the coefficient matrix to find the solution vector.
However, not all matrices have an inverse. A matrix is said to be singular if its determinant is zero, and it does not have an inverse. In such cases, we can use other methods, such as Gaussian elimination, to solve systems of linear equations.
Determinants Miscellaneous Exercise Solutions for Class 12th
The Miscellaneous Exercise in Determinants Class 12 NCERT Solutions 4, Determinants covers a wide range of topics related to determinants and their applications in linear algebra. This exercise includes various types of questions, such as finding the value of a determinant, using properties of determinants to simplify calculations, and solving systems of linear equations using Cramer’s rule.
The exercise begins by introducing the concept of the rank of a matrix, which is the maximum number of linearly independent rows or columns in the matrix. The rank of a matrix is essential in many applications of linear algebra, such as determining whether a system of linear equations has a unique solution or not.
The exercise also covers the concept of the inverse of a matrix and its properties, such as the product of the inverse of a matrix and the matrix itself equals the identity matrix. Students will learn how to find the inverse of a matrix and use it to solve systems of linear equations.
Another important topic covered in this exercise is the concept of eigenvalues and eigenvectors. Students will learn how to find the eigenvalues and eigenvectors of a matrix and how to use them to solve systems of linear equations.
The exercise also includes questions on the application of determinants in geometry, such as finding the area and volume of a geometric figure using determinants. Students will learn how to use determinants to calculate the area of a triangle, the volume of a parallelepiped, and the equation of a plane.
The Miscellaneous Exercise in Class 12 NCERT Solutions Chapter 4 on Determinants is a collection of questions that cover a wide range of topics related to determinants and their applications in linear algebra. This exercise includes various types of questions, such as finding the value of a determinant, using properties of determinants to simplify calculations, and solving systems of linear equations using Cramer’s rule.
The exercise begins with questions on the rank of a matrix, which is the maximum number of linearly independent rows or columns in the matrix. The rank of a matrix is essential in many applications of linear algebra, such as determining whether a system of linear equations has a unique solution or not.
Conclusion
Class 12 NCERT Solutions Chapter 4 on Determinants provides a comprehensive introduction to the fundamental concepts of determinants and their applications in linear algebra. Through the various exercises in this chapter, students learn how to calculate the value of a determinant, use properties of determinants to simplify calculations, find the inverse of a matrix, and apply determinants in geometry
Additionally, the chapter covers advanced topics such as eigenvalues and eigenvectors, which are essential in many areas of science and engineering. By mastering the concepts and techniques covered in this chapter, students can develop a strong foundation in linear algebra and prepare themselves for further studies in fields such as physics, engineering, and computer science.
The NCERT Solutions provided in this chapter offer step-by-step explanations and solutions to the exercises, making it easier for students to understand and apply the concepts covered in the chapter. Overall, this chapter provides a valuable resource for students seeking to gain a deeper understanding of determinants and their applications in linear algebra.
FAQ Related to Chapter 4 Determinants Class 12 NCERT Solutions 4 Free PDF
Frequently asked questions (FAQs) related to Chapter 4 Determinants Class 12 NCERT Solutions are a helpful resource for students who may have questions about the concepts covered in the chapter. Some of the commonly asked questions include the definition of determinants, how to find the determinant of a matrix, the importance of determinants, and the applications of determinants in real-life situations.
What are determinants?
Determinants are a mathematical concept used to determine various properties of square matrices, such as whether the matrix is invertible or not
What is the order of a matrix?
The order of a matrix is the number of rows and columns it has. For example, a matrix with 3 rows and 4 columns has an order of 3×4.
How do you find the determinant of a matrix?
The determinant of a matrix is computed using a specific formula that involves the elements of the matrix. The formula depends on the size of the matrix and can be calculated using cofactor expansion, row or column operations, or using the rule of Sarrus (for 3×3 matrices).
What is the importance of determinants?
Determinants have various applications in mathematics, physics, engineering, and economics. They are used to determine the properties of matrices, such as invertibility, rank, and eigenvalues. They are also used in solving systems of linear equations and in the study of linear transformations.
What is the adjoint of a matrix?
The adjoint of a matrix is a matrix obtained by taking the transpose of the matrix of cofactors of the original matrix. It is denoted by adj(A) or adjoint(A).
What is the inverse of a matrix?
The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. Not all matrices have an inverse, and the existence of an inverse is closely related to the determinant of the matrix.
What are some applications of determinants in real life?
Determinants have various applications in science, engineering, and economics. For example, they are used in physics to determine the eigenvalues of a quantum mechanical system. They are also used in engineering to solve systems of linear equations and in economics to determine the equilibrium price and quantity in a market.
