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Do you know how to solve a complex system of equations? How do search engines like Google list the websites one by one? Or how is the congested traffic flow managed in busy streets? The answers to all these questions can be found in Matrices, a powerful mathematical tool that can be used to solve complex systems of equations, also manage and manipulate data efficiently. From NCERT Class 12 Maths, the chapter Matrices, contains the Definition of a Matrix, Order of a Matrix, Types of Matrices, Operations on Matrices, Transpose of a Matrix, etc. Understanding these concepts will enable the students to solve matrix-related problems easily and also enhance their problem-solving ability in real-world applications.
This article on NCERT notes Class 12 Maths Chapter 3 Matrices offers well-structured NCERT notes to help the students grasp the concepts of Matrices easily. Students who want to revise the key topics of Matrices quickly will find this article very useful. It will also boost the exam preparation of the students by many folds. These notes of Class 12 Maths Chapter 3 Matrices are made by the Subject Matter Experts according to the latest CBSE syllabus, ensuring that students can grasp the basic concepts effectively. NCERT solutions for class 12 maths and NCERT solutions for other subjects and classes can be downloaded from the NCERT Solutions
A rectangular arrangement of objects (numbers or symbols, or any other objects) is called a matrix (plural: matrices).
Examples:
1.
2.
Rows and Columns:
The horizontal objects denote a row, and the vertical ones denote a column.
E.g., in the first matrix above, elements 2, 4 and -3 lie in the first row and 5, 4 and 6 in the second row.
Also, 2, 5 lie in the first column, 4,4 in the second column, and -3 and 6 in the third column.
Order of a matrix:
A matrix of order m × n (read as m by n matrix) means that the matrix has
E.g.,
The first matrix has order 2 x 3.
The second matrix has order 3 x 2.
The third matrix has order 4 x 1.
Representation of a
This representation can be represented in a more compact form as
Where
For example, to locate the entry in matrix A identified as aij, we look for the entry in row i, column j. In matrix A, shown below, the entry in row 2, column 3 is a23.
Note:
A matrix is only a representation of the symbol, number, or object. It does not have any value. Usually, a matrix is denoted by capital letters.
Row Matrix (Row vector) :
A matrix that has only one row.
Column Matrix (Column vector) :
A matrix that has only one column.
Zero or Null Matrix :
A matrix whose all entries are zero.
Square Matrix :
In a square matrix, the number of rows is equal to the number of columns.
Diagonal matrix:
A square matrix is said to be a diagonal matrix if all its elements except the diagonal elements are zero.
So, a matrix
E.g
A diagonal matrix of order n x n having diagonal elements as d1, d2, d3 ………, dn is denoted by
For example:
So, we can write
Scalar matrix:
A diagonal matrix whose all diagonal elements are equal is called a scalar matrix.
For a square matrix
Where c is not equal to 0
Unit or Identity Matrix:
A diagonal matrix of order n whose all diagonal elements are equal to one is called an identity matrix of order n. It is represented as I.
So, a square matrix
For example,
Matrices A and B are equal if '
(i) The order of matrix A is equal to the order of matrix B.
(ii)
If matrices A and B are of the same order, then the addition of matrices A and B is given by
A + B = [aij + bij]
Properties:
( a ) Addition of matrices is commutative : i.e. A + B = B + A
( b ) Matrix addition is associative : (A + B) + C = A + (B + C)
Additive inverse: Let two matrices A and B be such that A + B = O = B + A, then B is called the additive inverse of A.
Multiplication of a Matrix by a Scalar:
- If A and B are two matrices of the same order and 'k' is a scalar, then
k(A + B) = kA + kB.
- If p and q are two scalars and 'A' is a matrix, then
(p+q )A = pA + qA
- If p and q are two scalars and 'A' is a matrix, then
(pq )A =p (qA) =q (pA)
Let A be a matrix of order m x n and B be a matrix of order p x q, then the matrix multiplication AB is possible if and only if n = p. The multiplication of AB is given by
Properties of Matrix Multiplication:
(i) Matrix multiplication is not commutative: i.e.
( ii ) Matrix Multiplication Is Associative: If A, B & C are conformable for the product AB & BC, then (AB) C = A(BC)
( ii ) Matrix Multiplication Is Distributivity : A(B+C)=A B+A C and (A+B) C=A C+B C
Let
Properties of transpose :
Orthogonal Matrix:
A square matrix is said to be an orthogonal matrix if
Symmetric matrix :
A square matrix
Skew symmetric matrix :
A square matrix
Properties of symmetric & skew-symmetric matrices:
Matrix A is symmetric if
Matrix A is skew-symmetric if
The sum of two symmetric matrices is a symmetric matrix, and the sum of two skew-symmetric matrices is a skew-symmetric matrix.
If A & B are symmetric matrices, then AB + BA is a symmetric matrix and AB – BA is a skew-symmetric matrix.
Elementary operations of a matrix:
The following elementary operations are allowed on a matrix.
If A and B are two square matrices such that
Note:
Important points to note:
Happy learning !!!
Let be any matrix of order m x n then we can get hen transpose of matrix A by interchanging its rows with columns. i.e. AT or
.
No, CBSE doesn't provide any short notes or revision notes for any class.
NCERT textbook is the most important book for CBSE Class 12 Maths. You can also refer Maths books by RS Aggarwal and RD Sharma.
The total weightage of Matrices is 6 marks in the final board exam.
Some students consider Probability and Integration are the most difficult chapter in the CBSE Class 12 Maths but with rigorous practice, students can get command on these chapters very easily.
NCERT notes are helpful in understanding some important concepts and can be used to revise important concepts.
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