The Rank of a Matrix by Row- Echelon Form

The rank of a matrix in row echelon form can be determined by calculating the number of non-zero rows and non-zero columns. In the case of the row echelon form matrix, the first number in the row must be 1 and every 1 must be to the right of the above the number of it. The non-zero rows are above the ropes with zeros in the row echelon form. The first non-zero number is defined as the leading coefficient in the row echelon form of a matrix.

What is the Echelon Form of a Matrix?

In the row echelon form of a matrix, all zero is at the bottom and the first non-zero number must be to the right of the non-zero number in the above row. In the row echelon form, the rank of a matrix is the same for row rank and column rank. The rank of the matrix in the row echelon form is equal to the number of all non-zero rows. 

Above presented an example of a row echelon form matrix and the rank of the matrix can be calculated by determining the number of non-zero rows. Therefore, in the above matrix the number of non-zero rows is 3, hence the rank of the matrix is 4×3.

What are the Rules of Echelon Form?

• All zero rows must be at the bottom of the matrix in the two-echelon form
• The non-zero rows must enter  to the right of the previous row in the matrix
• The first number in the row must be 1 and the entry of the non-zero rows must be 1
• All non-zero rows must be above all-zero rows in the rectangular matrix

Explain the Row Echelon Form of a Matrix

In the row echelon, the leading coefficient of the non-zero rows is situated to the right position of the coefficient of the above row. It is essential to meet the condition of row echelon form that are in the column below the coefficient are zero. Generally, the leading coefficient is denoted by any number but in the case of row echelon form, the coefficient must be 1. Row echelon matrix is commonly associated with linear algebra where the properties of the matrix depend on the value of all non-zero rows.

Properties of a Matrix in Row Echelon Form

• The entry in the column as a form of non-zero value must be 1
• All entries in the column must be zero to the above or below a leading non-zero row
• Gaussian elimination can help in the transformation of the matrix in the row echelon form to determine the rank of the matrix

State the Conditions for Row Echelon Matrix

The primary condition for the row echelon form is that the number must be in the system of linear equations where the augmented matrix is transformed to the reduced row echelon form. The elementary row operation should be followed to transform any matrix to the row echelon form for the determination of the number of non-zero rows. All row echelon matrices and reduced row echelon matrices have the similar zero rows and indices are present in similar pivots.

Explain Reduced Row Echelon Form

A matrix in row echelon form can be transferred to the reduced row echelon form and the entry in the non-zero rows must be with 1. The value of the non-zero rows in the matrix helps to determine the position of the leading coefficient. Each column containing a non-zero number has zero in the other position to satisfy the condition of the row echelon form. Gauss Jordan elimination processes operate on the rank of the row and column in the reduced row echelon form.

Conclusion

The above study indicates that the row echelon form matrix can be transformed to the reduced row echelon form by using the Gauss Jordan elimination process. The rank of the matrix can be determined by calculating the number of non-zero rows in the row echelon form matrix. It is essential to start the first number as a 1 in the row of the row echelon matrix. The rank of a matrix is generally the dimension of a vector obtained by the column of the matrix.