# NCERT Solutions for Class 12 Maths Chapter 3 Free PDF Download

## NCERT Solutions for Class 12 Maths Chapter 3 Matrices

NCERT Solutions for Class 12 Maths Chapter 3 will strengthen studentâ€™s basic and conceptual fundamentals to score better marks and stay ahead. Our Mathematics experts who constantly work hard and keep detailed eyes on the subject have prepared these NCERT solutions.

NCERT Solutions for Class 12 Maths Chapter 3 Matrices is extremely helpful in revising complete syllabus and getting a strong base. NCERT solutions for Matrices Class 12 Maths give step by step solution for each and every question of the chapter. These solutions will also help you with your homework.

Our team of expert teachers has created NCERT solutions for class 12 Maths Chapter 3 Matrices according to curriculum and pattern of syllabus. Our app will help you to get complete NCERT solutions.

### CBSE Class 12 Maths Chapter 3 – Matrices NCERT Solutions

NCERT solutions for class 12 Maths Chapter 3 Matrices help students to understand matrices in an easy and self-explanatory way. The conceptual background of matrices is necessary for various branches of mathematics. Matrices are one of the most useful tools in mathematics as well as other areas of science like cryptography, genetics, economics, sociology, modern psychology and industrial management, etc.

### Sub-topics covered under NCERT Solutions for Class 12 Maths Chapter 3

• 3.1 Introduction
• 3.2 Matrix
• 3.2.1 Order of Matrix
• 3.3 Types of Matrices
• 3.4 Operations on Matrices
• 3.4.2 Multiplication of a matrix by a scalar
• 3.4.3 Properties of matrix addition
• 3.4.4 Properties of scalar multiplication of a matrix
• 3.4.5 Multiplication of matrices
• 3.4.6 Properties of multiplication of matrices
• 3.5 Transpose of a Matrix
• 3.5.1 Properties of the transpose of the matrices
• 3.6 Symmetric and Skew-Symmetric Matrices
• 3.7 Elementary Operation (Transformation) of a Matrix
• 3.8 Invertible Matrices
• 3.8.1 Inverse of a matrix by elementary operations

### NCERT Solutions for Class 12 Maths Chapter 3

Matrices deals with arranging and organizing them in a proper form of a table. In our NCERT Solutions for Class 12 Maths Chapter 3, students will also learn about various ways to apply matrices with determinants. This chapter covers all aspects of matrices with their types, operations, and applications.

Let us discuss the sub-topics in detail.

3.1 Introduction

It simplifies our work to a great extent when compared with other straight forward methods. Matrices are not only used as a representation of the coefficients in the system of linear equations but also used in a personal computer.

3.2 Matrix

A matrix is an ordered rectangular collection of numbers or functions. These numbers are called the elements or the entries of the matrix. It contains some Rows and Columns.

3.2.1 Order of Matrix

Each matrix has m rows and n columns. Its order is m Ã— n and we read it as an m by n matrix.

3.3 Types of Matrices

This topic discusses various types of Matrices as Column Matrix, Row Matrix, Square Matrix, Identity Matrix, etc.

3.4 Operations on Matrices

It introduces some basic but important operations on matrices, like the addition of matrices, multiplication of a matrix by a scalar, difference, and multiplication of matrices, etc. Also, it explains various properties of matrices based on these operations.

It elaborates the process to add two matrices.

3.4.2 Multiplication of a matrix by a scalar

It elaborates the process to multiply a matrix by a scalar quantity.

It shows some additive properties of matrices.

3.4.4 Properties of scalar multiplication of a matrix

It shows some multiplicative properties of matrices.

3.4.5 Multiplication of matrices

It elaborates the process to multiply two matrices.

3.4.6 Properties of multiplication of matrices

It shows some multiplicative properties of matrices.

3.5 Transpose of a Matrix

It explains the meaning and procedure of finding the transpose of a matrix. It plays the main role to define and find the symmetric and skew-symmetric matrices.

3.5.1 Properties of the transpose of the matrices

It explains some properties of the transpose of matrices.

3.6 Symmetric and Skew-Symmetric Matrices

It defines two more types of matrices which used to get some interesting results.

3.7 Elementary Operation (Transformation) of a Matrix

It explores the elementary transformations of the matrices. There are six transformations possible on a matrix, three of which are due to rows and three due to columns.

3.8 Invertible Matrices

It defines the existence of inverse of a matrix. This inverse matrix exists and gives the identity matrix after multiplying it with the original matrix, if it exists, it will be unique.

3.8.1 Inverse of a matrix by elementary operations

It discusses the elementary operations on the inverse of a matrix.

### Solved Questions for You

Question 1: MatricesÂ AÂ andÂ BÂ will be inverse of each other only if

1. AB=BA
2. AB=0,BA=I
3. AB=BA=0
4. AB=BA=I

Answer: We know that ifÂ AÂ is a square of orderÂ m, and if there exists another square matrixÂ BÂ of the same orderÂ m, such thatÂ AB=I, thenÂ BÂ is said to be the inverse ofÂ A.

In this case, it is clear thatÂ AÂ is the inverse ofÂ B.
Thus , matricesÂ AÂ andÂ BÂ will be inverses of each other only ifÂ AB=BA=I.

Question 2: If a matrix hasÂ 24Â elements, what are the possible order it can have? What, if it hasÂ 13Â elements?

Answer: We know that if a matrix is of the orderÂ mÃ—n, it hasÂ mnÂ elements.
ThusÂ to find all the possible orders of a matrix havingÂ 24Â elements, we haveÂ to find all the ordered pairs of natural numbers whose product isÂ 24.
The ordered pairs are(1,24),(24,1),(2,12),(12,2),(3,8),(8,3),(4,6)Â andÂ (6,4)
Hence, the possible orders of a matrix havingÂ 24Â elements are:
1Ã—24,24Ã—1,2Ã—12,12Ã—2,3Ã—8,8Ã—3,4Ã—6,6Ã—4
(1,13)Â andÂ (13,1)Â are the ordered pairs of natural numbers whose product isÂ 13.
Hence, the possible orders of a matrix havingÂ 13Â elements areÂ 1Ã—13Â andÂ 13Ã—1

Question 3: IfÂ n=p, then the order of the matrixÂ 7Xâˆ’5ZÂ is:Â

1. pÃ—2
2. 2Ã—n
3. nÃ—3
4. pÃ—n

Answer: In this, order ofÂ X=2Ã—n

and order ofÂ Z=2Ã—p

Therefore,Â n=p

Hence order ofÂ 7Xâˆ’5Z=2Ã—n.

Thus option (B) is correct.

Question 4: IfÂ A,BÂ are symmetric matrices of same order, thenÂ ABâˆ’BAÂ is a ,
A. Skew symmetric matrix

1. Symmetric matrix
2. Zero matrix
3. Identity matrix

Answer: Â Given,Â AÂ andÂ BÂ are symmetric matrices, therefore, we have:
Aâ€²=AÂ andÂ Bâ€²=B……….(i)
Consider
(ABâˆ’BA)â€²=(AB)â€²âˆ’(BA)â€²,[âˆµ(Aâˆ’B)â€²=Aâ€²âˆ’Bâ€²]
=Bâ€²Aâ€²âˆ’Aâ€²Bâ€²,[âˆµ(AB)â€²=Bâ€²Aâ€²]

=BAâˆ’ABÂ [by (i) ]

=âˆ’(ABâˆ’BA)
âˆ´(ABâˆ’BA)â€²=âˆ’(ABâˆ’BA)
Thus,Â (ABâˆ’BA)Â is a skew-symmetric matrix.

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NCERT Solutions for Class 12 Maths Chapter 3 Matrices is very popular among the students due to its high use in other application areas. These solutions will help students to frame a better understanding of it with a clear view of solving the problems.

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