bilqis*Eliminasi Gauss-Jordan menggunakan O.B.Eidem Gaussdisambung dengan :

* + =

* + =

* + =

baris 3 + baris 2baris 3 -2 + baris 1baris 2 -1 + baris 3

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bilqis*Suatu SPL mempunyai 3 kemungkinan jawaban, yaitu :1.Mempunyai jawaban tunggal2.Mempunyai banyak jawaban 3.Tidak mempunyai jawaban Contoh :Tentukan nilai a agar SPL berikut:

Mempunyai jawaban tunggalMempunyai banyak jawaban Tidak mempunyai jawaban

x 2y + 3z = 12x 3y + 9z = 4x 3y + (a2 - 4)z = 1 + a

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bilqis*Penyelesaian :

Matriks Eselon SPL di atas adalah : Mempunyai jawaban tunggala2 4 0 a -2 dan a 2

Mempunyai banyak jawaban a2 4 = 0 dan a +2 = 0 a = -2

iii.Tidak mempunyai jawaban a2 4 = 0 dan a + 2 0 a = 2

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bilqis*Lihat contoh di halaman 5 dan 6Lihat contoh di halaman 11 dan 12

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bilqis*Halaman 5Example 3. In the left column below we solve a system of equations by operating on the equations in the system, and in the right column we solve the same system by operating on the rows of the augmented matrix.

x + y + 2z = 92x + 4y 3z = 13x + 6y -5z = 0

Add -2 times the first equation to the second to obtainAdd -2 times the first row to the second to obtainx + y + 2z = 9 2y 7z = -173x + 6y -5z = 0

Add -3 times the first row to the third to obtainAdd -3 times the first equation to the third to obtainx + y + 2z = 9 2y 7z = -17 3y -11z = -27

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bilqis*Multiply the second equation by to obtainMultiply the second row by to obtainAdd -3 times the second equation to the third to obtainAdd -3 times the second row to the third to obtainMultiply the third equation by -2 to obtainMultiply the third row by -2 to obtain

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bilqis*Add -1 times the second equation to the first to obtainAdd -1 times the second row to the first to obtainAdd -11/2 times the third equation to the first and 7/2 times the third equation to the second to obtainAdd -11/2 times the third row to the first and 7/2 times the third row to the second to obtainThe solution : x = 1, y = 2, z = 3

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bilqis*Halaman 11

Step 1. Locate the leftmost column that does not consist entirely of zeros.

Step 2. Interchange the top row with another row, if necessary, to bring a nonzero entry to the top of the column found in Step 1.

Leftmost nonzero columnThe first and second rows in the preceding matrix were interchanged

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bilqis*Step 3if the entry that is now at the top of the coloumn found in step 1 is a, multiply the first row by 1/a in order to introduce a leading 1 step 4add suitable multiples of the top row to the rows below so that all entries below the leading 1 to zeros step 5Now cover the top row in the matrix and begin again with step 1 applied to the submatrix that remains. Continue in this way until the entire matrix is in row-echelon formThe first row of the preceding matrix was multiplied by

-2 times the first row of the preceding matrix was added to the third rowleft most nonzero coloumn in thesubmatrix

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Sheet1

12-53614

00-20712

0050-17-29

Sheet2

Sheet3

Sheet1

12-53614

00-20712

24-56-5-1

Sheet2

Sheet3

Sheet1

12-53614

00-20712

0050-17-29

Sheet2

Sheet3

bilqis*The first row in the submatrix was multiplied by -1/2 to introduce a leading 1-5 times the first row of the submatirx was added to the second row of the submatrix to introduce a zero below the leading 1

The top row in the submatrix wascovered, and we returned again to the step 1

The first(and only) row in the submatrix was multiplied by 2 to introduce a leading 1The entire matrix is now in row-echelonform. To find the reduce row-echelon form we need the following additional step

leftmost non zero coloumn in the new submatrix

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Sheet1

12-53614

0010-3.5-6

0050-1729

Sheet2

Sheet3

Sheet1

12-53614

0010-3.5-6

0050-1729

Sheet2

Sheet3

Sheet1

12-53614

0010-3.5-6

00000.51

Sheet2

Sheet3

Sheet1

12-53614

0010-3.5-6

000012

Sheet2

Sheet3

bilqis*Step 6Begining with the last nonzero row and working upward, add suitable multiplies of each row to the rows above to introduce zeros above the leading 1s7/2 times the third row of the preceding matrix was added to the second row-6 times the third row was added to the first row5 times the second row was added to the first row

The last matrix is in reduced row echelon form

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Sheet1

12-5302

001001

000001

Sheet2

Sheet3

Sheet1

12-53614

001001

000012

Sheet2

Sheet3

Sheet1

120307

001001

000012

Sheet2

Sheet3

bilqis*Sistem Persamaan Linier Homogen :Sistem Persamaan Linier dikatakan homogen jika semua suku di kanan tanda = adalah 0.Solusi Sistem Persamaan Linier Homogen:Solusi Trivial ( semua xi = 0; i = 1 .. n ): pasti ada Solusi Non-trivial ( solusi trivial, plus solusi di mana ada xi 0 )

Contoh: lihat contoh 6 halaman 18 dan verifikasi proses penyelesaiannya

22-1010 -1-12-31011-20-10001110

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bilqis*Contoh: lihat contoh 6 halaman 18 dan verifikasi proses penyelesaiannya

22-1010 -1-12-31011-20-10001110

11 -1/201/2 0-1-12-31 011 -20-1 000111 011 -1/201/2 000 3/2-33/2 000 -3/20-3/2 000111 0Brs-1 (1/2)Brs-2 + brs-1Brs-3 brs-1

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bilqis*11 -1/201/20003/2-33/2000 -3/20-3/200011 1011 -1/201/2 0001-21 000101 000111 011 -1/201/2 0001-21 000020 000030 0Brs-2 (2/3)Brs-3 ( 2/3)Brs-3 brs-2Brs-4 brs-2

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bilqis*11 -1/201/20001-21000020000030011 -1/201/20001-21000010000010011 -1/201/20001-210000100000000Brs-3 (1/2)Brs-4 (1/3)Brs-4 brs-3

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bilqis*11 -1/201/20001-21000010000000011 -1/201/2000101000010000000011 0010001010000100000000baris-1 + (1/2) baris-2

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bilqis*11 0010001010000100000000x1 + x2 + x5 = 0 x3 + x5 = 0 x4 = 0x5 = s x3 + x5 = 0 x3 = x5 x2 = t x1 + x2 + x5 = 0 x1 = x2 x5Ruang solusinya = { (-t-s, t, -s, 0, s ) }Catt => yang diumpamakan dahulu adalah index terbesar

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bilqis*Teorema:Sistem Persamaan Linier Homogen dengan variabel lebih banyak d/p. persamaan mempunyai tak berhingga banyak pemecahan.

Ditinjau dari matriksnya:Sistem Persamaan Linier Homogen dengan kolom lebih banyak d/p. baris mempunyai tak berhingga banyak pemecahan.

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bilqis*Contoh menggunakan MatlabSoalx + y + 2z = 92x + 4y 3z = 13x + 6y 5z = 0Buat matrix pada Matlab

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bilqis*MatlabMengenol-kan baris ke-2, kolom 1 Baris 2 = Baris 1 * -2 + baris 2

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bilqis*MatlabMengenol-kan baris ke-3, kolom 1 Baris 3 = Baris 1 * -3 + baris 3

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bilqis*MatlabMembuat nilai 1 pada kolom 2 dan baris 2 Baris 2 = Baris 2 * 1/2

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