Basic Elementary Row Operations of Matrix

• Elementary row operations are used to transform a system of linear equations into a new system that has the same solutions as the original one (i.e., into an equivalent system).

• There are three elementary operations:

1. Interchanging two rows.

2. Multiplying a row by a non-zero real number and replace in the same row.

3.  Adding a non-zero multiple of one row and replace that row.

Case 01: Interchanging two rows

 Any 2 rows of a matrix can be exchanged. If the ith and jth rows are exchanged, it is shown by Ri ↔ Rj 

                     Example:

Case 02: Multiplying a row by a non-zero real number and replace in the same row

The elements of any row of a matrix can be multiplied by a non-zero number. So if we multiply the ith row of a matrix by a non-zero number k, symbolically it can be denoted by Ri → kRi.

                    Example:-

Case 03: Adding a non-zero multiple of one row and replace that row

The elements of any row  can be added with the corresponding elements of another row  which is multiplied by a non-zero number. So if we add the ith row of a matrix to the jth row which is multiplied by a non-zero number k, symbolically it can be denoted by Ri → Ri + kRj.

                    Example:-

A matrix is in Row Echelon Form if it has the following properties:  

• Any row consisting entirely of zeros occurs at the bottom of the matrix.

• For each row that does not contain entirely zeros, the first non-zero entry is 1 (called a leading 1).

• Two successive rows are non-zero, then 2nd row must start with more zeros than the first row moving from left to right.

Examples:-

A matrix is in Reduced Row Echelon Form if it has the following properties:  

•  A matrix is in the reduced row echelon form if ;

·       The matrix in the row echelon form.

·       Any column containing a leading 1 has other elements zero.

  Example:-

Algorithm to convert a matrix into a row echelon form matrix:

Step 01:

• Find the 1st non-zero column moving from left to right & select a non – zero entry from that column. By interchanging rows (if necessary) ensure that the first entry in the column is non-zero. Apply the elementary row operations to set the first entry of that column as 1.

Step 02:

• Apply row operations of the type kR i+R j→Rj  for j>11 to get zeros under the first element of the first column obtained in the first step in each of the remaining row.

Step 03:

• Discard the first row and locate the next row that contains non-zero elements.  Apply row operations to get number 1 in to the second column of that row. Continue this process until the row echelon form is obtained.