# 1. Classification of Matrices, Definitions and its Important Properties

This article will cover about the classification of Matrices ,Definitions of various matrices and its properties .

Matrix: Systematic arrangement of M x N numbers in the form of M Rows and N Columns is said to be a matrix.

Notation:

Principle Diagram Elements:-

The element of a square matrix A such that i = j are said to be the principle diagonal elements.

Various types of Matrices which are useful for GATE Exam are explained below with examples

#### a.Upper Traingular Matrix:

a square matrix is said to be Upper Triangular Matrix if all the entries below the Main diagonal are zero.

#### B.Lower Triangular Matrix

A square matrix is said to be Lower triangular Matrix if all the entries above the main diagonal are zero.

#### C.Diagonal Matrix

For a square Matrix if all the elements except principal diagonal elements are zero then the matrix is known as Diagonal Matrix.

#### D.Scalar Matrix

The square matrix is said to be scalar matrix if it is having a constant value for all the elements of the principal diagonal and the other elements of the matrix are zero.

Note:

i. If all the elements of a Scalar Matrix are 1 then the matrix is said to be Identity Matrix or Unit matrix and it is denoted by (In)

ii. Constant X [Identity Matrix] = [Scalar Matrix]

#### E.Complex Matrix

If at least one of element of Matrix is Complex Number then the matrix is said to be Complex Matrix.

#### F.Conjugate Matrix

If each element of a Matrix is replaced with its Complex Conjugate Number then this matrix is said to be Conjugate matrix of the given matrix.

#### G.Hermition Matrix

If the Transpose of a congugate matrix is equal to the given matrix then it is said to be Hermition Matrix i.e.

#### H.Skew – Hermition Matrix

If the Transpose of a Congugate matrix is equal to the given matrix but with opposit sign, then the matrix is said to be Skew- Hermition Matrix i.e.

Note:

If A is Hermition matrix then, iA is Skew – Hermition Matrix and Vice – Versa.

#### I.Orthogonal Matrix

If the Inverse of matrix is equal to its Transpose, then the particular matix is said to be Orthogonal Marix i.e.

Note:

If AB = 0 then A (or) B need not be null matrix

## J.Null Matrix:

If all the elements of a matrix are zero, then the matrix is said to be a Null Matrix.

So far we have seen different types of Matrices and now we will discuss some important Properties of Matrices.

## PROPERTIES OF A MATRIX

a. Trace of a Matrix:

Trace of a matrix is defined only for square matrix and it is given as the sum of the elelments in principal diagonal.

Trace of a Matrix A is denoted as Tr(A) i.e.

Properties of Trace of a Matrix:

b. Traspose of a Matrix:

The transpose of a given matrix A is obtained by changing the rows into columns and columns into rows . It is denoted as AT

That means

Properties of Transpose of a Matrix:

Next article will cover the topics about Inverse of a Matrix,Co-factor of a Matrix,Adjoint of a Matrix,Minor of a Matrix and Rank of a Matrix

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