Definition

A Set of mn numbers ( real or imaginary ) arranged in the form of a rectangular array of m rows and n columns is called an m×n matrix ( to be read as 'm by n' matrix ).

An m×n matrix is usually written as -

m×n

Or   A = [ aij ]m×n

The numbers a11 ,a12, ….etc are known as the elements of the matrix A .

The element aij belongs to i^th row and j^th column and is called the ( i,j)^th element of the matrix .

For example

    A =     is a matrix having 2 rows and 3 columns and so it is a matrix of order 2×3 such

that a₁₁ =2 , a₁₂ =1 ,

 a₁₃ =-1 ,a₂₁ =1, a₂₂ =3 , a₂₃ =2 .

Types of Matrices

(1)Row Matrix

A matrix having only one row is called a row matrix.

For example--

        A = [ 1 2 -1 -2 ] is a row matrix of order 1×4.

(2) Column Matrix

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

For example --

   A =

is a column - matrix of order 3×1.

(3)Square Matrix

A matrix in which the number of rows is equal to the number of columns , say ‘n’ ,is called a square matrix of order n .

The elements a_ij of a square matrix A = [ a_ij ]_n×n for which i = j , that is , the elements a₁₁ ,a₂₂ ,a₃₃ , ……,a_nn are called the diagonal elements and the line along which they lie is called the principal diagonal or leading diagonal of the matrix .

For example

   A = 

is  a square matrix of order 3 in which the diagonal elements are  2,-2 and -3.

(4) Diagonal Matrix

A square matrix A = [ a_ij ]_n×n is called a diagonal matrix if all the elements , except those in the leading diagonal , are zero , that is  

          a_ij = 0 for all i ≠ j

For example , the matrix A = 

is a diagonal matrix.

(5)Scalar Matrix

A diagonal matrix in which all the diagonal elements are equal is called the scalar matrix .

For example , the matrix A = 

is a scalar matrix of order 3 .

(6)Identity or Unit Matrix

A square matrix each of whose diagonal element is unity and each of whose non-diagonal elements is equal to zero is called an identity or unit matrix .

The identity matrix of order n is denoted by I_n .

For example

The matrix I₂ = 

is an identity matrix of order 2.

(7)Null Matrix

A matrix whose all elements are zero is called a null matrix or a zero matrix.

For example -A =

is a null matrix of order 2×2.

(8)Upper Triangular Matrix

In an upper triangular matrix, all elements below the main diagonal are zero .

For example -

A = 

is an upper triangular matrix.

(9)Lower Triangular Matrix

In a lower triangular matrix, all elements above the main diagonal are zero.

For example--

A = 

 is a lower triangular matrix of order 3×3.