# Different Types of Matrices – Definitions and Examples

Here you will what is matrix and definitions of different types of matrices with examples.

Let’s begin –

## What is Matrix ?

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 $$\times$$ n matrix (to be read as m by n matrix).

A m by n matrix is usually written as

A = $$\begin{bmatrix}a_{11} & a_{12} & …… & a_{1n} \\ a_{21} & a_{22} & …… & a_{2n}\\ . & . & . \\ a_{m1} & a_{m2} & …… & a_{mn} \end{bmatrix}$$

## Different Types of Matrices

All different types of matrices with examples are given below :

#### Row Matrix

A matrix having only one row is called a row matrix or a row-vector.

Example : A = [ 1 2 -1 2 ] is a row matrix of order $$1 \times 4$$.

#### Column Matrix

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

Example : $$\begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}$$ is a column matrix of order $$3 \times 1$$.

#### Square Matrix

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

Example : the matrix $$\begin{bmatrix} 2 & 1 & -1 \\ 3 & -2 & 5 \\ 1 & 5 & -3 \end{bmatrix}$$ is a square matrix of order $$3 \times 3$$ in which diagonal elements are 2, -2 and -3.

#### Diagonal Matrix

A square matrix A = $$[a_{ij}]_{n\times n}$$ is called a diagonal matrix if all the elements, except those in the leading diagonal are zero.

Example : the matrix $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$ is a diagonal denoted by A = diag [1, 2, 3].

#### Scalar Matrix

A square matrix A = $$[a_{ij}]_{n\times n}$$ is called a scalar matrix if

(i) $$a_{ij}$$ = 0 for all i $$\ne$$ j and,

(ii) $$a_{ii}$$ = c, for all i, where c $$\ne$$ 0

In other words, a diagonal matrix in which all the diagonal elements are equal is called the scalar matrix.

Example : the matrix $$\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$$ is scalar martix of order 2.

#### Identity or Unit Matrix

A square matrix A = $$[a_{ij}]_{n\times n}$$ is called a identity or unit matrix if

(i) $$a_{ij}$$ = 0 for all i $$\ne$$ j and,

(ii) $$a_{ii}$$ = 1, for all i

In other words, a diagonal matrix in which all the diagonal elements is unity is called the unit matrix.

The identity matrix of order n is denoted by $$I_n$$.

Example : the matrix $$I_2$$ = $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ is identity matrix of order 2.

#### Null Matrix

A matrix in which al elements are zero is called  a null or a zero matrix,

Example : the matrix $$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$ is null matrix of order 2.

#### Upper Triangular Matrix

A square matrix A = $$[a_{ij}]$$ is called an upper triangular matrix if $$a_{ij}$$ = 0 for all  i > j.

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

Example : $$\begin{bmatrix} 1 & 3 & 4 \\ 0 & 4 & 5 \\ 0 & 0 & 7 \end{bmatrix}$$ is a upper triangular matrix.

#### Lower Triangular Matrix

A square matrix A = $$[a_{ij}]$$ is called an lower triangular matrix if $$a_{ij}$$ = 0 for all  i < j.

Thus, in an lower triangular matrix, all elements above the main diagonal are zero.

Example : $$\begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 1 & 6 & 5 \end{bmatrix}$$ is a lower triangular matrix.