Gaussian Elimination

If $(A^{\prime }|b^{\prime })$ obtained from $(A|b)$ by finite row operations, then $A^{\prime }x=b^{\prime }$ is equivalent to $Ax=b$

Gaussian elimination is where you use Elementary Matrix Operations on the augmented matrix to get it into reduced row echelon form

Reduced Row Echelon Form

• Requirements
  • Any row containing a nonzero entry precedes any row in which all the entries are zero (if any)
  • The first nonzero entry in each row is the only nonzero entry in its column
  • The first nonzero entry in each row is 1 and it occurs in a column to the right of the first nonzero entry in a preceding row
• The REF of a matrix is unique

Augmented Matrix

• $(A|B)$ is the augmented matrix of the system $Ax=b$
• Consistent $⟺\text{rank}(A)=\text{rank}(A|B)$