NCERT Class 12-Mathematics: Exemplar Chapter –3 Matrices Part 1

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3.1 Overview

3.1.1 A matrix is an ordered rectangular array of numbers (or functions). For example,

The numbers (or functions) are called the elements or the entries of the matrix.

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.

3.1.2 Order of a Matrix

A matrix having m rows and n columns is called a matrix of order or simply matrix (read as an m by n matrix).

In the above example, we have A as a matrix of order i.e., matrix.

In general, an m × n matrix has the following rectangular array:

The element, is an element lying in the row and column and is known as the element of A. The number of elements in an matrix will be equal to .

3.1.3 Types of Matrices

(i) A matrix is said to be a row matrix if it has only one row.

(ii) A matrix is said to be a column matrix if it has only one column.

(iii) A matrix in which the number of rows are equal to the number of columns, is said to be a square matrix. Thus, an matrix is said to be a square matrix if and is known as a square matrix of order ‘’.

(iv) A square matrix is said to be a diagonal matrix if it’s all non-diagonal elements are zero, that is a matrix is said to be a diagonal matrix if , when .

(v) A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix is said to be a scalar matrix if

, when

, when , for some constant .

(vi) A square matrix in which elements in the diagonal are all and rest are all zeroes is called an identity matrix.

In other words, the square matrix is an identity matrix, if , when and , when .

(vii) A matrix is said to be zero matrix or null matrix if all its elements are zeroes. We denote zero matrix by .

(ix) Two matrices and are said to be equal if

(a) They are of the same order, and

(b) Each element of A is equal to the corresponding element of B, that is, for all and .

3.1.4 Addition of Matrices

Two matrices can be added if they are of the same order.

3.1.5 Multiplication of Matrix by a Scalar

If is a matrix and k is a scalar, then is another matrix which is obtained by multiplying each element of A by a scalar k, i.e.

3.1.6 Negative of a Matrix

The negative of a matrix A is denoted by . We define .

3.1.7 Multiplication of Matrices

The multiplication of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B.

Let be an matrix and be an matrix. Then the product of the matrices A and B is the matrix C of order . To get the element of the matrix C, we take the row of A and column of B, multiply them element wise and take the sum of all these products i.e.,

The matrix is the product of A and B.

Notes:

1. If AB is defined, then BA need not be defined.

2. If A, B are, respectively matrices, then both AB and BA are defined if and only if and .

3. If AB and BA are both defined, it is not necessary that .

4. If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.

5. For three matrices A, B and C of the same order, if , then , but converse is not true.

6. , so on

3.1.8 Transpose of a Matrix

1. If be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A.

Transpose of the matrix A is denoted by A′ or (AT). In other words, if

, then .

2. Properties of transpose of the matrices

For any matrices A and B of suitable orders, we have

(i) ,

(ii) (where k is any constant)

(iii)

(iv)

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