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Section 2.2 Systems of Liner Equations: Unique Solutions The Gauss-Jordan Elimination Method Operations 1. Interchange any two equations. 2. Replace an equation by a nonzero constant multiple of itself. 3. Replace an equation by the sum of that equation and a constant multiple of any other equation. Ex. Solve the system 2x y z 3 step 1 x yz 2 2y 2 z 2 3y z 1 2 x y z 2 x 3 y 3z 0 x yz 2 y z 1 3 y z 1 Row 1 (r1) Row 2 (r2) Row 3 (r3) Replace r2 with [r1 + r2] Replace r3 with [–2(r1) + r3] Replace r2 with ½(r2) x yz 2 3 y z 1 2 z 4 Replace r3 with [–3(r2) + r3] x yz 2 4 y z 1 z 2 x y 5 y Replace r3 with ½(r3) 4 Replace r1 with [(–1)r3 + r1] 1 Replace r2 with [r2 + r3] z 2 3 x 6 y Replace r1 with [r2 + r1] 1 z 2 So the solution is (3, –1, –2) 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 System x y z 2 x 3 y 3z 0 2x y z 3 Augmented matrix 1 1 1 2 1 3 3 0 2 1 1 3 coefficients constants Row Operation Notation: 1.Interchange row i and row j Ri R j 2.Replace row j with c times row j cR j 3.Replace row i with the sum of row i and c times row j Ri cR j Ex. Last example revisited: Matrix System x y z 2 x 3 y 3z 0 1 1 1 2 1 3 3 0 2 1 1 3 2x y z 3 x yz 2 2y 2 z 2 3y z 1 R2 R1 R3 ( 2) R1 1 1 1 2 0 2 2 2 0 3 1 1 x yz 2 y z 1 1 R 2 2 3 y z 1 x yz 2 y z 1 2 z 4 R3 ( 3) R2 x yz 2 y z 1 z 2 1 R 2 3 1 1 1 2 0 1 1 1 0 3 1 1 1 1 1 2 0 1 1 1 0 0 2 4 1 1 1 2 0 1 1 1 0 0 1 2 x y y 4 1 R1 ( 1) R3 R2 R3 z 2 3 x y 1 R1 R2 z 2 This is in Row- Reduced Form 1 1 0 4 0 1 0 1 0 0 1 2 1 0 0 3 0 1 0 1 0 0 1 2 Row–Reduced Form of a Matrix 1. Each row consisting entirely of zeros lies below any other row with nonzero entries. 2. The first nonzero entry in each row is a 1. 3. 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. 4. If a column contains a leading 1, then the other entries in that column are zeros. Row–Reduced Form of a Matrix Row-Reduced Form Non Row-Reduced Form 1 0 0 3 0 1 0 1 0 0 1 2 1 0 0 9 0 0 1 4 0 1 0 2 1 0 0 8 0 1 0 4 0 0 0 0 R2 , R3 switched Must be 0 1 0 5 1 0 1 0 3 0 0 1 5 Unit Column A column in a coefficient matrix where one of the entries is 1 and the other entries are 0. 1 0 5 1 0 1 0 3 0 0 1 5 Unit columns Not a Unit column Pivoting – Using a coefficient to transform a column into a unit column 1 1 1 2 1 3 3 0 2 1 1 3 1 1 1 2 0 2 2 2 0 3 1 1 This is called pivoting on the 1 and it is circled to signify it is the pivot. Gauss-Jordan Elimination Method 1. Write the augmented matrix 2. Interchange rows, if necessary, to obtain a nonzero first entry. Pivot on this entry. 3. Interchange rows, if necessary, to obtain a nonzero second entry in the second row. Pivot on this entry. 4. Continue until in row-reduced form.