Echelon form is used to simplify a complex matrix into a simpler form by using row operations. A matrix is in echelon form if it has increasing numbers of zeros as the row number increases, and any rows of all zeros are at the bottom. For example, a 4x4 matrix is converted into echelon form by performing row operations to make the principal diagonal elements non-zero and elements below them zero, resulting in a matrix with rank 2 as indicated by the number of non-zero rows.

1. WELCOME

2. Echelon’s Form
By
Praneeth Naga Sai Narayan

3. Echelon’s Form:
 Echelon Form of a matrix is used to solve a complex matrix to a simple matrix.
 A matrix is in an Echelon Form if it satisfies some below given conditions:
 The number of zero’s are increased by the row number before the non- zero elements.
 If zero rows are exist that row is at the end of the matrix.
 Note: A given matrix can be converted into echelon’s form by using row operations only.

4. For Example:
Q)
𝟏 𝟐 𝟑 −𝟏
−𝟐 −𝟏 −𝟑 −𝟏
𝟏 𝟎 𝟏 𝟏
𝟎 𝟏 𝟏 𝟏
Ans:
𝟏 𝟐 𝟑 −𝟏
−𝟐 −𝟏 −𝟑 −𝟏
𝟏 𝟎 𝟏 𝟏
𝟎 𝟏 𝟏 𝟏
Principal Diagonal Elements
Note: You have to make all zero’s below the principal
diagonal elements by using row operation.
R2 = R2+2R1, R3 = R3-R1

5. 
𝟏 𝟐 𝟑 −𝟏
𝟎 𝟑 𝟑 −𝟑
𝟎 −𝟐 −𝟐 𝟐
𝟎 𝟏 𝟏 −𝟏
R3 = 3R3+2R2, R4 = 3R4-R2

𝟏 𝟐 𝟑 −𝟏
𝟎 𝟑 𝟑 −𝟑
𝟎 𝟎 𝟎 𝟎
𝟎 𝟎 𝟎 𝟎
∴ The above matrix is in the form of Echelon’s Form.
 Now, We have to find the rank of the matrix.
 To find the rank of the matrix, we have to consider
the non zero rows as the rank of matrix.
 Now, the rank of above matrix is 2.
∴ (𝐴) = 2