Archana

Nov. 6, 2019
**MATRIX**

Let us consider the following system of equations in three unknowns (variables) x, y and z.

x + y + z = 10

2x –z = – 3

3x + y + 2z = 17

If we arrange the coefficients of x, y and z in the order in which they occur in the given equations

and enclose them in brackets, we get the following arrangement:

Each arrangement of the kind given by the above representation of the system of Linear

equations is called a MATRIX. The numbers which constitute the matrix are called its entries or

elements.

**DEFINITION**

A matrix is not just a collection of elements, but it is a collection in which each array has a very

definite position i.e. it is an ordered array of elements.

OR

A set of mnelements arranged in the form of an ordered set of mrows, each row consisting of an

ordered set of n elements is called an m × n matrix. It is read as ‘m by n matrix”.**ELEMENTS OF A MATRIX**

The numbers occurring in the rectangular array (matrix) are called the elements of a matrix.

Each of the mn numbers of an m × n matrix is called an element of the matrix. The element

lying in the first row and first column is known as leading element.

**ORDER OF A MATRIX**

A matrix mby nis said to be of order m × n, if it has m rows and n columns.

REPRESENTATION OF A MATRIX

**MATRIX OVER A FIELD**

If all the elements of an m× nmatrix belong to a field F, then the matrix is said to be defined over

the field F. In particular, if all the elements of a matrix belong to the field of real numbers, then

the matrix is said to be real. The elements of the field F are called scalars.

DIAGONAL ELEMENTS OF A MATRIX

An element a ij of the matrix A = [a_{ij}]is said to be diagonal element if i = ji.e., the elements

a_{11},a_{22},a_{33} ……are the diagonal elements.**PRINCIPLE DIAGONAL**

The line along which the diagonal elements lie is called the principle diagonal or simply diagonal

of a matrix.**TYPES OF MATRICES****A. ROW MATRIX/ROW VECTOR**

A matrix having only one row is called a row matrix or a row matrix.**B. COLUMN MATRIX/COLUMN VECTOR**

A matrix having only one column is called a column matrix or column vector.

**C. RECTANGULAR MATRIX**

A matrix in which number of rows is not equal to the number of columns or vice-versa is called a

rectangular matrix.**D. SQUARE MATRIX**

A matrix having equal number of rows and columns is called a square matrix. If the matrix A has

n rows and n columns, it is said to be a square matrix of order n.**E. ZERO MATRIX/NULL MATRIX**

A matrix square or rectangular, each of whose element is zero is called a zero matrix or

null matrix. A zero matrix of order m × n is denoted by O m×n .**F. DIAGONAL MATRIX**

A square matrix A = [a ij ]is said to be a diagonal matrix, if all its elements except the diagonal

elements are zero. Thus a matrix A = [a ij ]is said to be diagonal matrix if