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Matrices

Row Reduction

Problems

Problems

Elementary Row Operations

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:

       


       

Row Reduction

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

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  6.    

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

   

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