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1. Rank
(By using normal & echelon form)
Amos Richard

2. Rank
• Rank is denoted by (‘r’)( ‘ρ’), Rank of a matrix of
independent rows or colums.
• Only null matrix or zero matrix has zero rank.
• For a matrix of order rank (ρ(A)) od ‘A’ is less than or
equal to minimum (m,n).
[A]mxn
ρ(A) ≤ min(m,n).
• For a non-singular matrix ‘A’ of order ‘n’
ρ(A)=n

3. Echelon Form Method For Rank
• We can transform a given non-zero matrix to a simplified form
called a Row-echelon form, using the row elementary operations.
• We may have rows all of whose entries are zero. Such rows are
called zero rows.
For example, consider the following matrix.

4. Here R1 and R2 are non-zero rows.
R3 is a zero row.
A non-zero matrix A is said to be in a row-echelon form if:
(i) All zero rows of A occur below every non-zero row of A.
(ii) The first non-zero element in any row i of A occurs in the jth
column of A, and then all other elements in the jth column of A
below the first non-zero element of row i are zeros.
(iii) The first on-zero entry in the ith row of A lies to the left of the
first non-zero entry in ( i + 1)th row of A.

6. Now we apply elementary transformations.
R2 → R2 – 2R1
R3 → R3 – 3R1
We get

7. R3 → R3 – 2R2
The above matrix is in row echelon
form.
Number of non-zero rows = 2
Hence the rank of matrix A = 2

8. Rank By Normal Form
• Any matrix ‘A’ can be reduce to the following forms .
A =
[IR] or [Ir 0]
• While ifinite number of elementory transformation these for
are called normal form
• Rank of the matrix by normal form is given by the order of
identity matrix

9. Rank of a Matrix Using Normal Form
Example:Find the rank of the matrix by reducing it to normal
form
Solution:

12. References
• https://www.rgpvonline.com
• https://byjus.com