NCERT Solutions for Class 12 Maths Chapter 3
12th Maths Chapter 3 Solutions in English Medium
|Class: 12||Maths (English and Hindi Medium)|
12th Maths Chapter 3 Solutions
Download NCERT Solutions Class 12 Maths chapter 3 exercise 3.1, 3.2, 3.3, 3.4 and miscellaneous exercises in PDF form English and Hindi Medium for CBSE Board. UP Board Students are using the same NCERT Books, so they can also use these solutions as UP Board Solutions 12 Maths Chapter 3 for the academic session 2021-22. We have done everything perfectly to provide solutions of Class 12 Maths for CBSE and UP Board, if you feel some problem, inform us. We will definitely rectify.
Class 12 Math Chapter 3 Solutions in Hindi Medium
Class 12 Maths Chapter 3 Solutions in PDF
Class 12 Maths Exercise 3.1 Video Solutions
Class 12 Maths Exercise 3.1 Question 1, 2, 3Class 12 Maths Exercise 3.1 Question 1, 2 in Hindi
Class 12 Maths Exercise 3.2 Video Solutions
Class 12 Maths Exercise 3.2 Question 1, 2Class 12 Maths Exercise 3.2 Question 1 in Hindi
Class 12 Maths Exercise 3.3 Video Solutions
Class 12 Maths Exercise 3.3 Question 1, 2Class 12 Maths Exercise 3.3 Question 1, 2 in Hindi
Class 12 Maths Exercise 3.4 Video Solutions
Class 12 Maths Exercise 3.4 Question 1Class 12 Maths Exercise 3.4 Question 1, 2 in Hindi
Class 12 Maths Miscellaneous Exercise 3 Video Solutions
Class 12 Maths Miscellaneous Exercise 3 Question 1Class 12 Maths Miscellaneous Exercise 3 Question 1 in Hindi
The arrangement of real numbers in a rectangular array enclosed in brackets as  or () is known as a Matrix(Matrices is plural of matrix). Matrix operations are used in electronic physics, computers, budgeting, cost estimation, analysis and experiments. They are also used in cryptography, modern psychology, genetics, industrial management etc. In general an m x n matrix is matrix having m rows and n columns. it can be written as follows:
Important Terms related to 12th Maths Chapter 3
Order of a Matrix
There may be any number of rows and any number of columns in a matrix. If there are m rows and n columns in matrix A, its order is m x n and it is read as an m x n matrix.
Transpose of a Matrix
The transpose of a given matrix A is formed by interchanging its rows and columns and is denoted by A’.
A square matrix A is said to be a symmetric matrix if A’ = A.
A square matrix A is said to be a skew symmetric if A’ = – A. all elements in the principal diagonal of a skew symmetric matrix are zeroes.
Addition of Matrix
If A and B are any two given matrices of the same order, then their sum is defined to be a matrix C whose respective elements are the sum of the corresponding elements of the matrices A and B and we write this as C = A + B.
Types of Matrices
- Row matrix: A row matrix has only one row but any number of columns.
- Column matrix: A column matrix has only one column but any number of rows.
- Square matrix: A square matrix has the number of column equal to the number of rows.
- Rectangular Matrix: A matrix is said to be a rectangular matrix if the number of rows is not equal to the number of columns.
- Diagonal matrix: If in a square matrix has all elements 0 except principal diagonal elements, it is called diagonal matrix.
- Scalar Matrix: A diagonal matrix is said to be a scalar matrix if all the elements in its principal diagonal are equal to some non-zero constant.
- Zero or Null matrix: If all elements of a matrix are zero, then the matrix is known as zero matrix and denoted by O.
- Unit or Identity matrix: If in a square matrix has all elements 0 and each diagonal elements are non-zero, it is called identity matrix and denoted by I.
- Equal Matrices: Two matrices are said to be equal if they are of the same order and if their corresponding elements are equal.
Properties of Matrices
- When a matrix is multiplied by a scalar, then each of its element is multiplied by the same scalar.
- If A and B are any two given matrices of the same order, then their sum is defined to be a matrix C whose respective elements are the sum of the corresponding elements of the matrices A and B and we write this as C = A + B.
- For any two matrices A and B of the same order, A + B = B + A. i.e. matrix addition is commutative.
- For any three matrices A, B and C of the same order, A + (B + C) = (A + B) + C i.e., matrix addition is associative.
- Additive identity is a zero matrix, which when added to a given matrix, gives the same given matrix, i.e., A + O = A = O + A.
- If A + B = O, then the matrix B is called the additive inverse of the matrix of A.
- If A and B are two matrices of order m x p and p x n respectively, then their product will be a matrix C of order m x n.
A square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = I = BA, Where I is identify matrix of order n.
Theorems of invertible matrices
- Theorem 1: Every invertible matrix possesses a unique inverse.
- Theorem 2: A square matrix is invertible iff it is non-singular.
Matrix is a latin word. Originally matrices are used for solutions of simultaneous linear equations. An important Chinese Text between 300 BC and 200 AD, nine chapters of Mathematical Art(Chiu Chang Suan Shu), give the use of matrix methods to solve simultaneous equations. Carl Friedrich Gauss(1777 – 1855) also gave the method to solve simultaneous equations by matrix method.
What is the matrix as per chapter 3 of class 12th Maths?
A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. We denote matrices by capital letters.
The following are some examples of matrices: –
In the above examples, the horizontal lines of elements are said to constitute, rows of the matrix and the vertical lines of elements are said to constitute, columns of the matrix. Thus A has 3 rows and 2 columns and B has 3 rows and 3 columns.
What are the practical implimentations of matrices chapter 3 of 12th Maths in different fields?
The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful tools in mathematics. The evolution of the concept of matrices is the result of an attempt to obtain compact and simple methods of solving a system of linear equations. Matrix notation and operations are used in electronic spreadsheet programs for personal computer, which in turn is used in different areas of business and science like budgeting, sales projection, cost estimation, analyzing the results of an experiment, etc.
Also, many physical operations such as magnification, rotation, and reflection through a plane can be represented mathematically by matrices. Matrices are also used in cryptography. This mathematical tool is not only used in certain branches of sciences but also in genetics, economics, sociology, modern psychology, and industrial management.
How many exercises are there in chapter 3 of class 12th Maths for 1st term exam?
There are 5 exercises in chapter 3 of class 12th Maths. The first exercise (Ex 3.1) has 15 sums (5 examples and 10 questions). The second exercise (Ex 3.2) contains 36 sums (14 examples and 22 questions). In the third exercise (Ex 3.3), there are 15 sums (3 examples and 12 questions). The fourth exercise has 21 sums (3 examples and 18 questions). There are 18 sums (3 examples and 15 questions) in the last (Miscellaneous) exercise.
Are there any theorems in chapter 3 of class 12th Maths?
Yes, there are four theorems in chapter 3 of class 12th Maths. All the theorems are nice, easy, and important. Proofs of these theorems are short and easy. Students can easily understand the theorems and proofs of these theorems.