**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 a_{11}, a_{22}, a_{33} ....... 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

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 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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