Def:

1. Sec 3.3 Reduced Row-Echelon Matrices Def: A matrix A in reduced-row-echelon form if • A is row-echelon form • All leading entries = 1 • A column containing a leading entry 1 has 0’s everywhere else

2. Gauss-Jordan Elimination Gauss-Jordan Elimination Gaussian Elimination

3. Echelon Matrix  Reduced Echelon Matrix • A  row-echelon form • Make All leading entries = 1 (by division) • Use each leading 1 to clear out any nonzero elements in its column • A  row-echelon form • Make All leading entries = 1 (by division) • Use each leading 1 to clear out any nonzero elements in its column

4. Leading variables and Free variables Free Variables

5. Leading variables and Free variables Example 3: Use Gauss-Jordan elimination to solve the linear system Solution: Gauss-Jordan

6. Reduced Echelon is Unique Theorem 1 : Every matrix is row equivalent to one and only one reduced echelon matrix NOTE: Every matrix is row equivalent to one and only one echelon matrix Row-equivalent Row-equivalent Echelon ReducedEchelon What is common

7. The Three Possibilities #unknowns =#equs Example #unknowns > #equs Square systm Example unique No sol. No sol.

8. Homogeneous System Homogeneous System NOTE: Every homog system has at least the trivial solution

9. Homogeneous System NOTE: Every homog system either has only the trivial solution or has infinitely many solutions HomogSystem INconsistent consistent 2 3 1 Unique Solution Infinitely many solutions No Solution Special case ( more variables than equations Theorem: Every homog system with more variables than equations has infinitely many solutions

10. Homogeneous System Theorem: Every homog system with more variables than equations has infinitely many solutions Homog #unknowns =#equs Example Homog #unknowns > #equs Square systm Example unique No sol. No sol.