This document defines and describes different types of matrices. It begins by defining a matrix as an arrangement of numbers, symbols or expressions in rows and columns. It then discusses the order of a matrix, elements within a matrix, and examples of 3x3 matrices. Several basic types of matrices are defined, including row matrices, column matrices, square matrices, rectangular matrices, diagonal matrices, null matrices, symmetric matrices, and skew-symmetric matrices. Related matrices such as the transpose, adjoint, and inverse of a matrix are also explained. The document concludes by defining the rank of a matrix and describing the properties of an echelon matrix.

1. MATRICES
BSM101
Name: Jay Prakash Kumar
Branch: CSE(AI)

2.  Definition:
A Matrix is arrangement of number,symbols or expression in column and row.
Order of Matrix:
(Number of rows X number of columns)
A system of of mn number arranged in a rectangular formation along m rows and n columns in a
brackets[ ] is called mxn Matrix.
A matrix is also denoted by a single capital letter.
 Elements in a matrix is arrenge ij position.
Example, 3X3 order of matrix
A = position of elements are
positions of elements
i=row, j=column. Position of r is ij,
where, i=3, j=3
a b c
f g h
p q r
11 12 13
21 22 23
31 32 33

3. TYPES OF MATRIX
SOME BASICS TYPES OF MATRICES
 ROW MATRIX: having only row elements
 COLUMN MATRIX: having only column elements.
 SQUARE MATRIX: whose number of rows and column is equal (m=n).
 RECTANGULAR MATRIX: whose column elements are not equal to row element.
 DIAGONAL MATRIX: A square matrix is called a diagonal matrix, if all diagonal
elements are non zero.
 NULL MATRIX: all the elements are zero.
 SYMMETRIC MATRIX: aij=aji for all i and j.
 SKEW-SYMMETRIC MATRIX: aij=-aji for all i and j (diagonal elements are zero).

4. RELATED MATRICES
 Transpose of a matrix: In a matrix interchange the rows and column to obtain
a matrix, is called transpose of a matrix.
 Example,
 A = is A’ =
 Adjoint of a square matrix: The determinant of square matrix is called the
adjoint of matrix
written as Adj.A.
The adjoint of A is the transposed matrix of cofactor of A.
1 6
2 5
3 9
6 8
1 2 3 6
6 5 9 8

5.  Inverse of a matrix: If A is a matrix, than a matrix B if it exists, such that AB=BA=I,
is called inverse of A.
Denoted by 𝐴−1
𝐴−1 =
𝐴 ⅆ𝑗𝐴
𝐴
.
 Ortohgonal matrix: when the product of matrix A and transpose of it is equal to
identity matrix.
A.A’=I
 Idempotent matrix: when the square of a matrix is equal to that matrix.
𝐴2
= 𝐴.
 Nilpotent matrix: when 𝐴𝑘 = 0 or null matrix.
Where k is a positive intiger.

6. RANK OF A MATRIX
 A MATRIX IS SAID TO BE OF RANK r WHEN,
(1) These exist at least one non-singular or square sub matrix of order r.
(2) Every square sub matrix of order more than r is always singular.
denoted by p(A).
Example,
A = there is a minor order of 2, which is not zero.
p(A) = 2.
ECHELON MATRIX
 A MATRIX IS SAID TO BE ECELON MATRIX
(1) All zero rows of a follow all non-zero matrix.
(2) The number of zeros before the first non-zero element of a row increases as we pass from row
to row downwards.
1 3
5 9

7. THANK YOU.