Rank of a Matrix in Echelon Form with an example.pptx

Rank of a matrix in Echelon form using Elementary Transformation

1. Rank of a Matrix in Echelon Form MCA204B13 : Mathematical Foundations for Computer Science Dr. Soya Mathew Assistant Professor in Mathematics Department of Physical Sciences Kristu Jayanti College (Autonomous) Bengaluru – 560077, India

2. Matrix: Definition A set of mn numbers (real or imaginary) arranged in the form of rectangular array of m rows, each row consisting of an ordered set of n numbers is called an matrix. (read as m by n). An matrix is usually represented as A or where  

3. Rank of a Matrix The rank of a matrix is a unique real number associated to the given matrix. It is a very useful and has its application in solving the system of linear equations. Definition: Let be a non-zero matrix. The rank of is defined as the maximum of the orders of the non-singular square sub-matrices of . The rank of a null matrix is defined as zero. The rank of the matrix is denoted as or .  

4. Elementary Transformations: Let be the given matrix. An elementary transformation is either of the following: Interchange of any two rows (or columns). The symbol used is . Multiplication of the elements of a row (or column) by a non-zero number. The symbol used is . Adding to the elements of a row (or column), times the corresponding elements of another row (or column) where is a non-zero number. The symbol used is .  

5. Equivalent Matrices: Two matrices and are said to be equivalent if one matrix can be obtained from the other matrix by performing a finite number of elementary operations on . It is denoted by . Note: Equivalent matrices have same rank. Rank of a matrix does not alter by interchanging any two rows of the matrix. Rank of a matrix does not alter by multiplying elements of a row by a non-zero scalar. Rank of a matrix and the rank of its transpose are same.  

6. Echelon Form: A matrix is said to be in the echelon form if the number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row. Steps to reduce to echelon form Make the leading element (first non-zero entry) of each row as unity. Make all the entries below this leading entry as zero. The number of zeros appearing below the leading entry in each row is greater than that appears in its previous row. The zero rows must appears below the non – zero rows.

7. Finding the rank of a matrix using elementary transformations: Using elementary row transformations every matrix can be reduced to echelon form. Then the number of non – zero rows in echelon form of matrix is the rank of . Examples: , Here , Here  

8. Example: Find the rank of the matrix A using elementary row operation where Solution: Given ,  

9. , ,  

10. Thank You !!!