Row reducing allows you to systematically manipulate a matrix into reduced row echelon form using elementary row operations. These operations include swapping two rows, multiplying a row by a non-zero constant, and adding a multiple of one row to another. The document provides an example of using row operations like choosing pivots to get zeros below and above the pivot entries. It also shows expressing the solution of a system of equations in terms of parameters when there are more unknowns than equations.

1. Row Reducing How to get to Reduced Row Echelon Form

3. Get the leading 1’s starting at the pivot

7. EXAMPLE I PIVOT Use this one, called the pivot, to get zeros below. -3R1 + R2-> R2 R1 + R3-> R3

8. EXAMPLE I -3R1 + R2-> R2

9. EXAMPLE I R1 + R3-> R3

10. EXAMPLE I New pivot location We want a 1 here but we also want to avoid fractions -2R3 + R2-> R2

11. EXAMPLE I New pivot Use this new pivot to get zeros above and below R2 + R1-> R1 -2R2 + R3-> R3

12. EXAMPLE I New pivot location Now get a 1 here. R3-> R3

13. EXAMPLE I New pivot Use this new pivot to make zeros above. 11R3 + R1-> R1 13R3 + R2-> R2

14. EXAMPLE I

15. EXAMPLE II Solve the system: There are two equations and four unknowns so we need a 2 x 5 matrix. Augmented matrix:

16. EXAMPLE II -2R1 + R2-> R2 R2-> R2 ←RREF

17. EXAMPLE II Expressing the solution: If a system is consistent, any column that does not contain a leading one will have a parameter. Parameters required for x3 and x4

18. EXAMPLE II Expressing the solution: LET s t

19. EXAMPLE II Converting back to equations in terms of s and t: Solving for x1 and x2 leads to the final solution: