Maths NCERT Solutions For Class 12 Chapter 3 Matrices Exercise Free PDF

Get the complete Maths NCERT Solutions for Class 12 Chapter 3 Matrices in this article. Today I am also going to provide a PDF of all the exercises of the matrix. Learn the basic concepts and techniques of matrices to solve problems and excel in your exams.

Here is a table of Maths NCERT Solutions for Class 12 Chapter 3, including Exercise 3.1, Exercise 3.2, Exercise 3.3, Exercise 3.4, and the Miscellaneous Exercise on matrices:

Along with the NCERT textbook for Class 12 Maths, you will also find all the solutions in PDF here today. It is important to practice these exercises and understand the concepts thoroughly to do well in the subject.

In this article, we will explore the NCERT Maths Solutions for Class 12 Chapter 3: Matrices in detail. We will cover the important concepts, techniques, and formulas related to matrices, as well as provide some helpful tips and tricks to help you excel in your exams.

Get Free PDF Maths NCERT Solutions For Class 12 Chapter 3 Exercise 3.1

Class 12 Maths Matrices Exercise 3.1 is an important part of the Maths syllabus. This exercise deals with matrices and covers various topics such as matrix addition, subtraction, multiplication, identity matrix, inverse matrix, and transpose of a matrix. The exercise is designed to help students understand the fundamental concepts of matrices and provide them with a clear understanding of the subject.

What is a Matrix?

Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. They are used to represent data and perform operations like addition, subtraction, and multiplication. In this chapter, students will learn about the different types of matrices, their properties, and their operations.

For example, a 2×3 matrix (A) might look like this:

| 1 2 3 |
| 4 5 6 |

Here, the matrix has 2 rows and 3 columns, and the numbers in the matrix are called its entries. Matrices can be added, subtracted, and multiplied together, and there are many operations and techniques for manipulating matrices that are used in a wide range of mathematical and practical applications.

There are several types of matrices commonly used in mathematics, each with its own properties and applications. Some of the most common types of matrices include:

You Can Also Read: NCERT Maths Solutions For Class 12 Chapter 2

Types of Matrics:

1. Square matrix: A matrix with an equal number of rows and columns is called a square matrix. For example, a 3×3 matrix is a square matrix because it has three rows and three columns.
2. Diagonal matrix: A matrix where all the non-diagonal entries are zero is called a diagonal matrix. The diagonal entries can be any number, including zero.
3. Identity matrix: A square matrix with all diagonal entries equal to 1 and all non-diagonal entries equal to 0 is called an identity matrix. It is denoted by “I”.
4. Upper triangular matrix: A square matrix in which all entries below the main diagonal are zero is called an upper triangular matrix.
5. Lower triangular matrix: A square matrix in which all entries above the main diagonal are zero is called a lower triangular matrix.
6. Symmetric matrix: A square matrix that is equal to its own transpose is called a symmetric matrix. That is, A = A^T.
7. Skew-symmetric matrix: A square matrix that is equal to the negative of its transpose is called a skew-symmetric matrix. That is, A = -A^T.

These are just a few of the many types of matrices used in mathematics. Each type of matrix has its own unique properties and applications in various fields.

Overall, Class 12 Maths Matrices Exercise 3.1 is an important part of the Maths syllabus and is designed to provide students with a comprehensive understanding of matrices. By practising the exercise, students can improve their understanding of the subject and perform well in their exams.

Maths NCERT Solutions For Class 12 Chapter 3 Exercise 3.2 Matrix

Maths NCERT Solutions for Class 12 Chapter 3 Exercise 3.2 is an important topic that deals with the multiplication of matrices. This exercise is a part of the CBSE Class 12 Maths curriculum and is designed to help students develop their problem-solving skills.

Exercise 3.2 includes a set of problems that require students to use the properties of matrix multiplication to solve them. The exercise covers topics such as distributive property, associative property, and identity matrix, which are fundamental concepts of matrix algebra.

The Maths NCERT Solutions for Class 12 Chapter 3 Exercise 3.2 are designed to provide step-by-step solutions to all the problems in the exercise. These solutions are written in simple language and are easy to understand. They are also provided with diagrams and illustrations to help students visualize the problems.

Matrix addition involves adding corresponding elements of two matrices. To add two matrices, they must have the same dimensions. The sum of two matrices is another matrix of the same dimensions, where each element is the sum of the corresponding elements of the original matrices.

Matrix subtraction is similar to matrix addition, except that the corresponding elements of one matrix are subtracted from the corresponding elements of another matrix. As with addition, the matrices being subtracted must have the same dimensions.

Matrix multiplication is a more complex operation. To multiply two matrices, the number of columns of the first matrix must be equal to the number of rows of the second matrix. The product of two matrices is another matrix with a number of rows equal to the number of rows of the first matrix and a number of columns equal to the number of columns of the second matrix. The multiplication process involves multiplying the elements of the first matrix with the corresponding elements of the second matrix and then adding the products.

Matrix multiplication has several properties, including the distributive property, associative property, and identity matrix. The distributive property states that the multiplication of a matrix by a sum of matrices is equal to the sum of the multiplication of the matrix by each of the matrices. The associative property states that the multiplication of three matrices can be done in any order and the result will be the same. The identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere. When a matrix is multiplied by the identity matrix, the result is the original matrix.

Maths NCERT Solutions For Class 12 Chapter 3 Exercise 3.3

Maths NCERT Solutions For Class 12 Chapter 3 Exercise 3.2 covers the topic of matrix multiplication, which is a fundamental concept in linear algebra. In this exercise, we will explore the properties of matrix multiplication and learn how to find the product of two matrices.

Dear students Maths NCERT Solutions For Class 12 Chapter 3 Exercise 3.3 is also very important. In this, we learn to multiply two matrices. You will get the solution to all its questions in PDF which you can see.

Matrix multiplication is a way of combining two matrices into a new matrix by multiplying their corresponding elements and summing the products.

To multiply two matrices A and B, the number of columns in A must be equal to the number of rows in B. If A is an m x n matrix and B is an n x p matrix, then their product AB is an m x p matrix.

The element at position (i,j) in the product matrix AB is obtained by multiplying the elements of the i^th row of A with the elements of the j^th column of B and summing the products:

AB(i,j) = A(i,1) * B(1,j) + A(i,2) * B(2,j) + … + A(i,n) * B(n,j)

Matrix multiplication is not commutative, which means that AB and BA are not necessarily equal. However, matrix multiplication is associative, which means that (AB)C = A(BC) for any matrices A, B, and C of appropriate sizes.

Matrix multiplication also distributes over addition, which means that A(B + C) = AB + AC and (A + B)C = AC + BC for any matrices A, B, and C of appropriate sizes.

Finally, in the identity matrix I have the property that AI = IA = A for any matrix A of appropriate size.

Now we know that Maths NCERT Solutions For Class 12 Chapter 3 Exercise 3.4 Matrix

Let us now know about Maths NCERT Solutions For Class 12 Chapter 3 matrix ex 3.4. Exercise 3.4 also has an important question from which questions come in the exam. In this, we will learn about the transpose of a matrix and some of its properties.

The transpose of a matrix A is obtained by flipping its rows and columns. The transpose of A is denoted by A^T.

For example, if A is a matrix with elements a_ij, then its transpose A^T has elements a_ji.

The transpose of a matrix has several properties:

1. (A^T)^T = A, which means that the transpose of a matrix is the original matrix.
2. (A + B)^T = A^T + B^T, which means that the transpose of the sum of two matrices is equal to the sum of their transposes.
3. (kA)^T = kA^T, where k is a scalar, which means that the transpose of a scalar multiplied by a matrix is equal to the scalar multiplied by the transpose of the matrix.
4. (AB)^T = B^T A^T, which means that the transpose of the product of two matrices is equal to the product of their transposes in reverse order.
5. If A is a symmetric matrix, then A^T = A, which means that the transpose of a symmetric matrix is equal to itself.
6. If A is an invertible matrix, then (A^T)^-1 = (A^-1)^T, which means that the inverse of the transpose of an invertible matrix is equal to the transpose of the inverse of the matrix.

These properties make the transpose of a matrix a useful tool in linear algebra and matrix algebra. For example, the transpose can be used to simplify calculations involving dot products or to convert row operations to column operations and vice versa.

Maths NCERT Solutions For Class 12 Chapter 3 Miscellaneous Exercise of Matrix

We will now learn about Class 12 Chapter 3 Miscellaneous Exercise of Matrix because this is also a very important part of Chapter 3. Dear students, I would like to tell you that Maths NCERT Solutions For Class 12 Chapter 3 Matrices also has a variety of question papers, which creates a lot of questions for your exam. Let us also know something about the various questionnaires.

Class 12 Chapter 3 Miscellaneous Exercise of Matrix covers a wide range of topics related to matrices, including operations such as addition, subtraction, multiplication, and transpose. The exercises in this chapter require a thorough understanding of matrix operations and their properties.

One of the key concepts in this chapter is matrix multiplication. The product of two matrices involves multiplying their corresponding elements and summing the products. It is important to note that matrix multiplication is not commutative, meaning that the order in which matrices are multiplied matters.

Another important concept covered in this chapter is the transpose of a matrix. The transpose of a matrix is obtained by interchanging its rows and columns. The transpose has several important properties, such as (A^T)^T = A and (AB)^T = B^T A^T.

In the Miscellaneous Exercise, students are asked to solve various problems that require them to apply their understanding of matrix operations and their properties. These problems include finding the inverse of a matrix, solving systems of linear equations using matrices, and determining the rank of a matrix.

One common problem in this chapter involves finding the determinant of a matrix. The determinant of a matrix is a scalar value that can be used to determine whether the matrix is invertible and to find the inverse of the matrix if it exists. The determinant can be found using various methods, including cofactor expansion and row reduction.

Another important topic covered in this chapter is eigenvalues and eigenvectors. An eigenvalue is a scalar value that is associated with a matrix, and an eigenvector is a non-zero vector that satisfies a specific equation involving the matrix and the eigenvalue. Eigenvalues and eigenvectors have numerous applications in various fields, including physics, engineering, and computer science.

In other words, you can understand that Class 12 Chapter 3 Miscellaneous Exercise of Matrix covers a broad range of topics related to matrices, including operations such as multiplication, addition, subtraction, and transpose. Students who understand these concepts and their properties will be able to solve a wide range of problems, including finding the inverse of a matrix, solving systems of linear equations, and determining the rank of a matrix. The chapter also introduces important topics such as determinants and eigenvalues, which have numerous applications in various fields.

Conclusion: Maths NCERT Solutions For Class 12 Chapter 3 provides a comprehensive introduction to the concept of matrices, including operations such as addition, subtraction, multiplication, and transpose. The chapter also covers important topics such as determinants, the inverse of a matrix, the rank of a matrix, and eigenvalues and eigenvectors.

The chapter begins with an introduction to matrices and their basic operations. Students are then introduced to matrix multiplication, which is a fundamental concept in matrix algebra. The chapter also covers the transpose of a matrix and its properties, which are essential for solving problems in linear algebra.

One of the key highlights of this chapter is the introduction of determinants, which are important scalar values associated with matrices. Determinants are used to determine whether a matrix is invertible and to find the inverse of a matrix if it exists. The chapter covers various methods for finding determinants, including cofactor expansion and row reduction.

The chapter also covers the concept of the rank of a matrix, which is a measure of the number of linearly independent rows or columns in a matrix. The rank of a matrix is useful for determining the consistency and uniqueness of solutions to systems of linear equations.

Another important topic covered in this chapter is eigenvalues and eigenvectors, which have numerous applications in various fields, including physics, engineering, and computer science.

Overall, Maths NCERT Solutions For Class 12 Chapter 3 provides a thorough introduction to matrices and their operations, including important concepts such as determinants, rank, and eigenvalues. The chapter provides students with a solid foundation in linear algebra, which is essential for advanced studies in mathematics, engineering, physics, and other fields.