Table of Contents

## What is Matrix?

In Mathematics, a Matrix (plural Matrices) is a rectangular array or table of integers organized in rows and columns. The numbers are referred to as the matrix’s elements or entries. A matrix’s size is determined by the number of rows and columns it has.

## Application Area of Matrix

Matrices are used in a variety of fields, including Graph theory, Analysis and Geometry, Probability Theory and Statistics, Symmetries and Transformations in Physics, Linear Combinations of Quantum States, and Electronics.

## Order of the Matrix

If there are m rows and n columns, the matrix is said to be an “m by n” matrix, written “m × n.” For example,

m is row number means there are 2 rows and n is column number so there are 3 columns. So m x n = 2 x 3.

Here 1 2 3 and 4 5 6 are rows (See horizontally). 1 4, 2 5, 3 6 are columns (See vertically).

Let us take another example.

As you can notice m=3 and n=3 so Order of the matrix is 3 x 3. You can also say, this is a square matrix as its no of rows and columns are equal.

Here 4 5 6, 1 3 5, 2 6 1 are rows. 4 1 2, 5 3 6, 6 5 1 are columns

## Types of Matrices

### Row Matrix

A matrix is said to be row matrix if it has one and only one row. For example, A = [ 4 5 9 ]

### Column Matrix

A matrix is said to be column matrix if it has one and only one column. For example, here A is a column matrix

### Null or Zero Matrix

A matrix is said to be Null or Zero matrix if all the elements of matrix are zero.

### Square Matrix

A matrix is n x n matrix, the matrix having the same number of rows as columns. Here A is 2 x 2 square matrix

### Diagonal Matrix

A square matrix in which all the elements except its principle diagonal are zero, is known as Diagonal Matrix. Here A is 3 x 3 Diagonal matrix.

### Scalar Matrix

A Diagonal matrix in which elements of its principle diagonal are equal, is known as Scalar Matrix. Here A is 3 x 3 scalar matrix, in which its diagonal elements are equal (10).

### Identity or Unit Matrix

A Scalar matrix in which elements of its principle diagonal are 1 (one), is known as Identity or Unit Matrix. Here I is 3 x 3 identity matrix.

### Triangular Matrix

#### Upper Triangular Matrix

A square matrix, in which all the elements below principle diagonal are zero, is known as Upper Triangular Matrix.

#### Lower Triangular Matrix

A square matrix, in which all the elements above principle diagonal are zero, is known as Lower Triangular Matrix.

### Transpose of Matrix

The transpose of a matrix is found by interchanging its rows into columns or columns into rows. If “A” is the given matrix, then the transpose of the matrix is represented by A’ or A^{T}. If matrix A is of order 3 x 2 then A’ is of order 2 x 3. In below example, i just interchanged the rows into columns (or columns into rows).

### Symmetric Matrix

A square matrix A is said to be asymmetric matrix , if A = A’.

### Skew Symmetric Matrix

A square matrix A is said to be asymmetric matrix , if A = – A’.

### Singular Matrix

A square matrix A is said to be singular matrix , if | A | = 0. (determinant A is equal 0).

### Idempotent Matrix

Idempotent matrix is a square matrix which when multiplied by itself, gives back the same matrix. A matrix is said to be an idempotent matrix if A^{2} = A. Every identity matrix can be termed as an idempotent matrix.

### Nilpotent Matrix

A square matrix *A* is called a nilpotent matrix, if A^{n}=0 for some positive integer n.

Here A & B both are nilpotent matrix.

### Orthogonal Matrix

A square matrix A, such that A×A^{T}= I, is called an orthogonal matrix, where *I* is an identity matrix and A^{T} is the transpose of matrix A.

Here if we do A x A^{T} = I & B X B^{T} = I

## Equal Matrix

A matrix , say A = [a_{ij}] is said to be equal to matrix B = [b_{ij}], means A = B, if A & B have same order and a_{ij} = b_{ij} for each i & j. Here A & B both are equal matrix as each element of matrix A is exactly at same position in matrix B.