Matrices ISC Class-12 Maths ML Aggarwal Solutions Ch-3

Matrices ISC Class-12 Maths ML Aggarwal Solutions Ch-3. Step by step Solutions of ML Aggarwal ISC Mathematics Class-12  Exercise 3.1, Exe 3.2 And Exe 3.3, Exe 3.4, Exe 3.5 With Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

Matrices ISC Class-12 Maths ML Aggarwal Solutions Ch-3

Board	ISC
Class	12
Subject	Mathematics
Chapter-3	Matrices
Session	2024-25
Topics	Solutions of ML Aggarwal

Matrices

Matrices Definition: A matrix is a function which consists of an ordered rectangular array of numbers. The numbers in the array are called the entities or the elements of the matrix. The horizontal array of elements in the matrix is called rows, and the vertical array of elements are called the columns. If a matrix has m rows and n columns, then it is known as the matrix of order m x n.

Types of Matrices

Depending upon the order and elements, matrices are classified as:

Column Matrix: A column matrix is a m × 1 matrix, consisting of a single column of m elements. It is also called a column vector.

Row Matrix: A row matrix is a 1 × m matrix, consisting of a single row of m elements. It is also called a row vector.

Square Matrix: A matrix which has an equal number of rows and columns. It is expressed as m × m.

Diagonal Matrix: A matrix which has non-zero elements in its diagonal part running from upper left to the lower right or vice versa.

Scalar Matrix:  The scalar matrix is a square matrix, which has all its diagonal elements equal and all the off-diagonal elements as zero.

Identity Matrix: A square matrix which has all its principal diagonal elements as 1’s and all non-diagonal elements as zeros.

Zero Matrix: A matrix whose all entries are zero. It is also called a null matrix.

Transpose of a Matrix

If A = [a_ij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A and is denoted by A′ or (A^T ).

In other words, if A = [a_ij] m × n , then A′ = [a_ji] n × m

Equality of Matrices

Two matrices are said to be equal if-

(i) The order of both the matrices are the same

(ii) Each element of one matrix is equal to the corresponding element of the other matrix

Symmetric and Skew Symmetric Matrices

A square matrix A = [a_ij] is said to be symmetric if the transpose of A is equal to A, that is, [a_ij] = [a_ji] for all possible values of i and j.

A square matrix A = [a_ij] is a skew-symmetric matrix if A′ = – A, that is a_ji = – a_ij for all possible values of i and j. Also, if we substitute i = j, we have a_ii = – a_ii and thus, 2a_ii = 0 or a_ii = 0 for all i’s. Therefore, all the diagonal elements of a skew-symmetric matrix are zero.

Elementary Operation (Transformation) of a Matrix

There are six operations (transformations) on a matrix, three of which are due to rows, and three are due to columns, known as elementary operations or transformations.

1. The interchange of any two rows or two columns.
2. The multiplication of the elements of any row or column by a non zero number.
3. The addition to the elements of any row or column, the corresponding elements of any other row or column are multiplied by any non zero number.

Invertible Matrices

Suppose a square matrix A of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A, and it is denoted by A^-1. Also, matrix A is said to be an invertible matrix here

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Exe-3.1

Matrices ISC Class-12 Maths ML Aggarwal Solutions Ch-3

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Exe-3.2

Matrices ISC Class-12 Maths Solutions Ch-3

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Exe-3.3

Matrices ISC Class-12 Maths Solutions Ch-3

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Exe-3.4

Matrices ISC Class-12

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Exe-3.5

Matrices ISC Class-12 Maths Solutions Ch-3

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Chapter Test

Matrices ISC Class-12 Maths ML Aggarwal Solutions Ch-3

 -: End of Matrices ISC Class-12 Maths ML Aggarwal Solutions Ch-3:-

Return to :-  ML Aggarwal Solutions for ISC Class 12 Maths Vol-1

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