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We have said that we would like to perform elementary operations in our matrix to get simpler ones. But what does simpler mean? Which is the ideal situation we should pursue?


A matrix is said to be in row echelon form if

  1. All zero rows (rows containing only $0$) are below the nonzero ones
  2. The first nonzero coefficient in a nonzero row (called leading coefficient or pivot) is $1$
  3. Each pivot is strictly to the right of the pivot of the row above it

    $$ \left( \begin{array}{ccccccccc} 1 & \times & \times & \times & \times & \times & \times & \times & \times \\ \cdot & \cdot & 1 & \times & \times & \times & \times & \times & \times \\ \cdot & \cdot & \cdot & 1 & \times & \times & \times & \times & \times \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 1 & \times & \times \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 1 & \times \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \end{array} \right) $$

A matrix is said to be in reduced row echelon form if moreover

  1. All coefficients above a pivot are $0$

$$ \left( \begin{array}{ccccccccc} 1 & \times & \cdot & \cdot & \times & \times & \cdot & \cdot & \times \\ \cdot & \cdot & 1 & \cdot & \times & \times & \cdot & \cdot & \times \\ \cdot & \cdot & \cdot & 1 & \times & \times & \cdot & \cdot & \times \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 1 & \cdot & \times \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 1 & \times \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \end{array} \right) $$