1  Elementary Row Operations and Equivalent Systems of Linear Equations

Given a system of linear equations

a 1 , 1 x 1 + a 1 , 2 x 2 + + a 1 , n x n = b 1 a 2 , 1 x 1 + a 2 , 2 x 2 + + a 2 , n x n = b 2 a m , 1 x 1 + a m , 2 x 2 + + a m , n x n = b m (1.1)

We know that the following three types of operations lead to an equivalent system of linear equations:

  1. Interchange two equations.
  2. Multiply an equation by a nonzero number.
  3. Add a multiple of one equation to another equation.

These operations have their counterpart in operations with matrices.

The (matrix) elementary row operations are:

  1. Interchange two rows.
  2. Multiply a row by a nonzero real number.
  3. Add a multiple of one row to another row.

We have the following proposition:

Proposition 1.1. Given a system of linear equations. Denote by AR the augmented matrix of the system. Let C be a matrix that was obtained from the matrix ARby one or more elementary row operations. Then C is the augmented matrix of an equivalent system of linear equations.