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 AT. 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 A2 = A. Every identity matrix can be termed as an idempotent matrix.
Nilpotent Matrix
A square matrix A is called a nilpotent matrix, if An=0 for some positive integer n.
Here A & B both are nilpotent matrix.
Orthogonal Matrix
A square matrix A, such that A×AT= I, is called an orthogonal matrix, where I is an identity matrix and AT is the transpose of matrix A.
Here if we do A x AT = I & B X BT = I
Equal Matrix
A matrix , say A = [aij] is said to be equal to matrix B = [bij], means A = B, if A & B have same order and aij = bij 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.
