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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 Replace r2 with [r1 + r2] Replace r3 with [–2(r1) + r3] Replace r2 with ½(r2) 1 2 Row 1 (r1) Row 2 (r2) Row 3 (r3) step
4 5 3 Replace r3 with [–3(r2) + r3] Replace r3 with ½(r3) Replace r2 with [r2 + r3] Replace r1 with [( –1) r3 + r1]
So the solution is (3, –1, –2) 6 Replace r1 with [r2 + r1]
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 Augmented matrix coefficientsconstants
Row Operation Notation: 1.Interchange row i and row j 2.Replace row j with c times row j 3.Replace row i with the sum of row i and c times row j
Ex. Last example revisited: System Matrix
This is in Row- Reduced Form
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 FormNon Row-Reduced Form R 2, R 3 switched Must be 0
Unit Column A column in a coefficient matrix where one of the entries is 1 and the other entries are 0. Unit columnsNot a Unit column
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.
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.