(Reduced) Row Echelon Form

A matrix is in row echelon form if:

• The first non-zero element in each row, called the leading entry, is 1.
• Each leading entry is in a column to the right of the leading entry in the previous row.
• Rows with all zero elements, if any, are below rows having a non-zero element.
• For a non-zero row, the leading entry in the row is the only non-zero entry in its column(Reduced Echelon Form).

Solutions of $AX=b$ when $A$ is in Reduced row echelon form.

• Let $AX=b$ be a system of linear equations and suppose $A$ is in reduced row echelon form.
• Suppose for some $i$, the $i^{th}$ row of $A$ is a zero but $b_i\neq 0$. Then this system has no solution.

Reason: This means if we write the corresponding system of linear equations, the $i^{th}$ equation reads, $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0x_1+0x_2+....0x_n=b_i$ Since $b_i\neq0$ , this equation cannot be satisfied.

• Assume that for every zero row of $A$, the corresponding entry of $b$ is also 0.
  • If the $i^{th}$ column has the leading entry of some row, we call $x_i$ a dependent variable.
  • If the $i^{th}$ column does not have the leading entry of some row, we call $x_i$ an independent variable.
• Assign arbitrary values to independent variables.
• For a dependent variable, there is a unique equation in which it occurs.
• All other variables in that equation are independent variables and thus have values assigned.
• Hence, we can compute the value of the dependent variable from this equation substituting the assigned values for the other independent variables in the equation.
• The obtained values for $x_i$ give a solution to the system.