Matematika Pertemuan 20 Matakuliah: D0024/Matematika Industri II Tahun : 2008

Bina Nusantara Sistem Persamaan Linier (SPL) A matrix that has undergone Gaussian elimination is said to be in row echelon form or, more properly, "reduced echelon form" or "row-reduced echelon form." Such a matrix has the following characteristics:Gaussian elimination 1. All zero rows are at the bottom of the matrix 2. The leading entry of each nonzero row after the first occurs to the right of the leading entry of the previous row. 3. The leading entry in any nonzero row is All entries in the column above and below a leading 1 are zero. Another common definition of echelon form only requires zeros below the leading ones, while the above definition also requires them above the leading ones. Echelon Form

Bina Nusantara Gaussian elimination is a method for solving matrix equations of the formmatrix equationsof the form To perform Gaussian elimination starting with the system of equations Gaussian Elimination

Bina Nusantara compose the "augmented matrix equation" (3) (3) Here, the column vector in the variables is carried along for labeling the matrix rows. Now, perform elementary row operations to put the augmented matrix into the upper triangular formcolumn vectorelementary row operationsupper triangular

Bina Nusantara Solve the equation of the th row for, then substitute back into the equation of the st row to obtain a solution for, etc., according to the formula For example, consider the matrix equationmatrix equation

Bina Nusantara In augmented form, this becomes (7) (7) Switching the first and third rows (without switching the elements in the right-hand column vector) gives (8) (8) Subtracting 9 times the first row from the third row gives

Bina Nusantara Subtracting 4 times the first row from the second row gives (10) (10) Finally, adding times the second row to the third row gives (11) (11) Restoring the transformed matrix equation gives (12) (12) which can be solved immediately to give, back- substituting to obtain (which actually follows trivially in this example), and then again back-substituting to find

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