Matakuliah
Tahun
: D0024/Matematika Industri II
: 2008
Matematika
Pertemuan 20
Sistem Persamaan Linier (SPL)
Echelon Form
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:
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 1.
4. 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.
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Gaussian Elimination
Gaussian elimination is a method for solving matrix
equations of the form
To perform Gaussian elimination starting with the
system of equations
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compose the "augmented matrix equation"
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 form
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(
3
)
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 equation
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In augmented form, this becomes
(
7
)
Switching the first and third rows (without switching the elements in
the right-hand column vector) gives
(
8
)
Subtracting 9 times the first row from the third row gives
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Subtracting 4 times the first row from the second row gives
Finally, adding
times
the second row to the third
row gives
(
1
1
)
Restoring the transformed matrix equation gives
(
1
2
)
which can be solved immediately to give
, backsubstituting to obtain
(which actually follows trivially in this
example), and then again back-substituting to find
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(
1
0
)
• Kerjakan latihan pada modul soal
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