Row Reduction and Systems of Linear Equations

One main benefit of matrices and vectors is the notational ease which their theory offers. Explicitly written out systems of m linear equations in n unknowns xt have the form 

Systems of linear equations can be simplified without affecting their solution by 

These three legitimate operations relate to row operations performed on the augmented matrix 

The row operations (a) to (c) are performed on (A b) until the front m by n matrix A achieves row echelon form. In a row echelon form R of A each row has a first nonzero entry, called a pivot, that is further to the right than the leading nonzero entry (pivot) of any previous row, or it is the zero row. 

The particular upper triangular shape of R makes the equation corresponding to its last nonzero row have the least number of variables so that it can be solved most easily. The remaining equations are then solved from the REF via backsubstitution from the bottom row on up. 

---