2. Elementary Row Operations
• Row Replacement:
– row = row + multiple of another row
• Row Interchange
– Switch two rows
• Scaling
– Multiply all entries of a row by a constant (≠0)
• Row operations are reversible
• Matrices row equivalent same soln set
3. Row Echelon Form (REF)
• All nonzero rows above any rows of all zeros
• Each leading entry (left-most non-zero entry –
PIVOT position) of a row is in a column to the
right of the leading entry above it
• All entries in a column below a leading entry
are zeros
• REF is NOT unique
4. Reduced Row Echelon Form (RREF)
• Leading entry in each non-zero row is 1
• Each leading entry is only non-zero entry in its
column
• RREF is unique
5. Row Reduction Algorithm
• Step 1: Left most nonzero column is a pivot
column. Pivot position is at top of column
• Step 2: Select a nonzero entry in the pivot column
as a pivot. If necessary, interchange rows to move
this entry into the pivot position
• Step 3: Use row replacement to create zeros in all
positions below the pivot.
• Step 4: Cover (ignore) the row containing the
pivot & any rows above it. Appy steps 1-3 to
submatrix that remains.
6. Row Reduction Algorithm - Backward
• To go to RREF:
• Step 5: Begin with rightmost pivot & work
backward (to the left) and upward, using row
replacement to create zeros in above each
pivot. If a pivot is not 1, make it 1 by a scaling
operation
7. Solutions to Linear Systems
• No solution inconsistent
• Single solution consistent
• Many solutions consistent
– General solution basic vars in terms of free vars
8. Existence & Uniqueness Thereom
• A linear system is consistent iff the rightmost
column of the augmented matrix is not a pivot
column, i.e., iff the REF form of the
augmented matrix has no rows of the form:
– [0 0 0 … 0 b]
10. Process for Solving Systems of Equns
1. Write augmented matrix
2. Reduce to REF
– Consistent?
3. Reduce to RREF consistent?
4. Write system corresponding to 3
5. Solve equations for basic vars in terms of free
variables.
no
Yes – keep going
