Rank of a Matrix

Elementary operations associated with a matrix

Following are the elementary row transformations (can be applied for columns also)

If matrix A gets transferred into another matrix B by applying any of the above transformation, then A is said to be equivalent to B,(A~B)

Echelon form

A matrix A of order m×n is said to be in a row reduced echelon form if

1. The leading element (the first non-zero entry )of each row is unity .

2. All entries below this leading entry is zero.

3. The number of zero’s appearing before the leading entry in each row is greater than that appearing in its previous row.

4. The zeros must appear below the non-zero rows.


Normal form or canonical form of the matrix

1. The given matrix A is reduced to an echelon form first by applying a series of elementary row transformations.

2. Later column transformations are performed to reduce the matrix to one of the following four forms,called normal form of A.

where Ir=Identity matrix.

Rank of a matrix

The Rank of the matrix A in its echelon form is equal to the number of non-zero rows .It is denoted by ρ(A).

Steps to find the rank of the matrix

1. To reduce the given matrix to a row echelon form, the first entry in the first row or the leading element should be non-zero, much preferably unity.

2. If the first entry is zero then we can interchange with any suitable row to meet the requirement.

3. Then by performing suitable row operations for the entire row, we have to make all the other elements in that column as zero (starting from the first row).

4. For the obtained row echelon form, we can write the rank of the matrix by counting the number of non-zero rows.


  1. Rank of a Matrix
  2. Row Operations

Related Problems

  1. Find the rank of the matrix A=[(2,1,3,5),(4,2,1,3),(8,4,7,13),(8,4,-3,-1)] by reducing it to echelon form
  2. Find the rank of the matrix [(1,1,1,6),(1,-1,2,5),(3,1,1,8),(2,-2,3,7)]
  3. Find the rank of the matrix A=[(0,1,-3,-1),(1,0,1,1),(3,1,0,2),(1,1,-2,0)] by reducing it to echelon form
  4. Find the rank of the matrix [(4,0,2,1),(2,1,3,4),(2,3,4,7),(2,3,1,4)]
  5. Find the rank of the matrix [(2,-1,-3,-1),(1,2,3,-1),(1,0,1,1),(0,1,1,-1)]



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