Gauss-Jordan Method for Solving Systems of Linear Equations

1. Section 2.2 Systems of Liner Equations: Unique Solutions

2. The Gauss-Jordan Elimination Method Operations • Interchange any two equations. • Replace an equation by a nonzero constant multiple of itself. • Replace an equation by the sum of that equation and a constant multiple of any other equation.

3. Row 1 (r1) Row 2 (r2) Row 3 (r3) Ex. Solve the system step 1 Replace r2 with [r1 + r2] Replace r3 with [–2(r1) + r3] 2 Replace r2 with ½(r2)

4. 3 Replace r3 with [–3(r2) + r3] 4 Replace r3 with ½(r3) Replace r1 with [(–1)r3 + r1] 5 Replace r2 with [r2 + r3]

5. Replace r1 with [r2 + r1] 6 So the solution is (3, –1, –2)

6. Augmented Matrix *Notice that the variables in the preceding example merely keep the coefficients in line. This can also be accomplished using a matrix. A matrix is a rectangular array of numbers Augmented matrix System coefficients constants

7. Row Operation Notation: • Interchange row i and row j • Replace row j with c times row j • Replace row i with the sum of row i and c times row j

8. Ex. Last example revisited: Matrix System

10. This is in Row- Reduced Form

11. Row–Reduced Form of a Matrix • Each row consisting entirely of zeros lies below any other row with nonzero entries. • The first nonzero entry in each row is a 1. • In any two consecutive (nonzero) rows, the leading 1 in the lower row is to the right of the leading 1 in the upper row. • If a column contains a leading 1, then the other entries in that column are zeros.

12. Row–Reduced Form of a Matrix Row-Reduced Form Non Row-Reduced Form R2 , R3 switched Must be 0

13. Unit Column A column in a coefficient matrix where one of the entries is 1 and the other entries are 0. Unit columns Not a Unit column

14. Pivoting – Using a coefficient to transform a column into a unit column This is called pivoting on the 1 and it is circled to signify it is the pivot.

15. Gauss-Jordan Elimination Method • Write the augmented matrix • Interchange rows, if necessary, to obtain a nonzero first entry. Pivot on this entry. • Interchange rows, if necessary, to obtain a nonzero second entry in the second row. Pivot on this entry. • Continue until in row-reduced form.