TYPES OF MATRICES

Row Matrix :

A matrix is said to be a row matrix, if it has only one row. A row matrix is also called as a row vector.

For example,

A  =  (5   3   4   1) and B  =  (–3   0   5 )

are row matrices of orders 1 x 4 and 1 x 3 respectively.

In general,

A  =  (a_ij)_1xn

is a row matrix of order 1xn.

Column Matrix :

A matrix is said to be a column matrix, if it has only one column. It is also called as a column vector.

For example,

are column matrices of orders 2x1 and 3x1 respectively.

In general,

A  =  [a_ij]_mx1

is a row matrix of order mx1.

Square Matrix :

A matrix in which the number of rows and the number of columns are equal is said to be a square matrix.

For example,

are square matrices of orders 2 and 3 respectively.

In general,

A  =  [a_ij]_mxm

is a square matrix of order m.

The elements a₁₁, a₂₂, a₃₃ ....... a_mm  are called principal or leading diagonal elements of the square matrix A.

Diagonal Matrix :

A square matrix in which all the elements above and below the leading diagonal are equal to zero, is called a diagonal matrix.

For example,

are diagonal matrices of orders 2 and 3 respectively.

In general,

A  =  [a_ij] _mxm 

said to be a diagonal matrix if a_ij  =  0 for all i ≠ j.

Note :

Some of the leading diagonal elements of a diagonal matrix may be zero.

Scalar Matrix :

A diagonal matrix in which all the elements along the leading diagonal are equal to a non-zero constant is called a scalar matrix.

For example,

are scalar matrices of orders 2 and 3 respectively.

In general,

A  =  [a_ij] _mxm

is said to be a scalar matrix if

a_ij = 0, when i ≠ j

a_ij = k. when i = j

where k is constant.

Unit Matrix :

A diagonal matrix in which all the leading diagonal entries are 1 is called a unit matrix. A unit matrix of order n is denoted by I_n. For example,

are unit matrices of orders 2 and 3 respectively.

In general, a square matrix A = [a_ij] _nxn is a unit matrix if

a_ij = 1 if i = j

a_ij = 0 if i ≠ j

A unit matrix is also called an identity matrix with respect to multiplication. Every unit matrix is clearly a scalar matrix. However a scalar matrix need not be a unit  matrix. A unit matrix plays the role of the number 1 in numbers.

Null Matrix or Zero-Matrix :

A matrix is said to be a null matrix or zero-matrix if each of its elements is zero. It is denoted by O.

For example,

are null matrices of order 2x3 and 2x2.

(i) A zero-matrix need not be a square matrix. 

(ii) Zero-matrix plays the role of the number zero in numbers.

(iii) A matrix does not change if the zero-matrix of same order is added to it or subtracted from it.

Transpose of a Matrix :

Definition The transpose of a matrix A is obtained by interchanging rows and columns of the matrix A and it is denoted by A^T (read as A transpose).

For example,

In general, if A  =  [a_ij] _mxn then A^T  =  [b_ij]_nxm where

b_ij  =  a_ij

for i  =  1, 2,......n and j  =  1, 2,...... m.

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