Def: A matrix A in reduced-row-echelon form if 1)A is row-echelon form 2)All leading entries = 1 3)A column containing a leading entry 1 has 0’s everywhere else Sec 3.3 Reduced Row-Echelon Matrices

Gauss-Jordan Elimination Gaussian Elimination

1)A  row-echelon form 2)Make All leading entries = 1 (by division) 3)Use each leading 1 to clear out any nonzero elements in its column Echelon Matrix  Reduced Echelon Matrix 1)A  row-echelon form 2)Make All leading entries = 1 (by division) 3)Use each leading 1 to clear out any nonzero elements in its column

Leading variables and Free variables Free Variables

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

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 Echelon Reduced Echelon Row-equivalent What is common

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

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

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

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