We can perform three elementary row operations on matrices:

• Multiplying a row by a constant.
• Switching two rows.
• Adding a constant times a row to another row.

Examples:

• Multiplying a row by a constant:
  
  
  

• Switching two rows:
  
  
  
  
  
  
  

• Adding a constant times a row to another row:
  
  
  
  
  
  
  

We perform row operations to row reduce a matrix; that is, to convert the matrix into a matrix where the first m×m entries form the identity matrix:

where * represents any number.

This form is called reduced row-echelon form.

Note: Reduced row-echelon form does not always produce the identity matrix, as you will learn in higher algebra. For our purposes, however, we will consider reduced row-echelon form as only the form in which the first m×m entries form the identity matrix.

To row reduce a matrix:

1. Perform elementary row operations to yield a "1" in the first row, first column.
2. Create zeros in all the rows of the first column except the first row by adding the first row times a constant to each other row.
3. Perform elementary row operations to yield a "1" in the second row, second column.
4. Create zeros in all the rows of the second column except the second row by adding the second row times a constant to each other row.
5. Perform elementary row operations to yield a "1" in the third row, third column.
6. Create zeros in all the rows of the third column except the third row by adding the third row times a constant to each other row.
7. Continue this process until the first m×m entries form the identity matrix.

Example: Row reduce

1. 
   

2. 
   

3. 
   

4. 
   

5. 
   

6. 
   

This is the reduced row-echelon form of the matrix: