The Rank of a Matrix Solved Example

The rank of a matrix is denoted by the number of rows and columns that are independent of each other. Therefore, to determine the rank of a matrix, the matrix needs to transform to the echelon form to identify the number of non-zero rows in the matrix. In the echelon form of the matrix, the entry of the row must be 1 and all the bottom numbers must be zero. In the echelon form of the matrix, the entry of the non-zero number must be to the right of the leading coefficient in the above row.

Define the Rank of a Matrix

The rank of a matrix can be determined by calculating the total number of non-zero rows in the row echelon form. For example,

A= 

In the above matrix, the second row is double the first row. The matrix has a total of two rows that are not in count as there is no zero at the bottom of the matrix. Therefore the rank of the above matrix is 1. 

How to Write the Rank of a Matrix?

The rank of the matrix is determined by calculating the number of non-zero rows in the echelon matrix. A unit matrix can be transformed into the row echelon matrix by using the row elementary formula. The rank of a matrix is determined by following the rules of the row echelon matrix. The rank of the matrix cannot be greater than the number of rows and columns in the united matrix. Calculation of the rank of the matrix depends on the properties of the row echelon matrix.

The Rank of a Matrix Solved Example

Let A = 

The rank of the above matrix can be determined by the following calculation:

From the rule of the elementary row operation, R2→R2-2R1 and R3→R3-3R1

Hence, the above matrix transformed to 

Secondly based on the formula R3→R3-2R2 the above matrix transformed to the below form:

The above matrix is the row echelon matrix and the number of non-zero rows is two. Hence the rank of the matrix is denoted by A= 2.

How to Determine the Rank of a Matrix?

Let A= 

The above matrix is the row echelon matrix as all the button numbers are zero and the first element of the row is 1. Here in this matrix, the first and second rows are the non-zero rows and the third row is not a non-zero row. Therefore A is a non-zero matrix as all the zero rows are below the non-zero rows in the matrix. Therefore, the rank of the above matrix is 2×3 as there are two non-zero rows.

The Rank of the Matrix for Reduced Row Echelon Form

The rank of the matrix is equal to the number of non-zero rows in the reduced row echelon form and it is denoted as A= r ≤ m. The rank of the matrix depends on the Gauss elimination method in the operation of row and column to transform to the row echelon form. The maximum number of the linear independence row in the matrix is defined as the row rank. The maximum number of linearly independent columns is denied as the column rank. 

Conditions for Row Echelon Matrix

• The first integer in the non-zero rows must be 1 and all the zero must be in the bottom
• The non-zero integers must be situated in the right of the coefficient of the nonzero integer in the above row
• All the non-zero items must be above the zero rows in the bottom
• The rank of the row cannot be greater than  the number of rows and column

Conclusion

The above study shows that the rank of the matrix depends on the value of non-zero rows in the row echelon matrix. The row echelon matrix is transformed from the unit matrix by using the row elementary formation. Calculating the number of rows and columns also helps to determine the process of Gauss elimination. It is essential to convert all the nonzero integers in the bottom rope to zero to determine the rank of the matrix.