Terminology
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Augmented Matrix (расширеннаяматрица): |
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матрицы( |
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Coefficient Matrix (матрица коэффициентов): |
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entry |
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элемент |
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6
List of operations to eliminate unknowns:
1.Multiply an equation through by a nonzero constant.
2.Interchange two equations.
3.Add a multiple of one equation to another.
The corresponding operations on the rows of the augmented matrix (elementary row operations):
1.Multiply a row through by a nonzero constant.
2.Interchange two rows.
3.Add a multiple of one row to another row.
7
EXAMPLE
x2x3x
+y +2z =9
+4y −3z =1
+6y −5z = 0
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Add |
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−3 |
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• -2 x the 1 |
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row to the 2 |
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• -3 x the 1st row to the 3rd |
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−5 |
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Multiply the 2nd row |
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by 1/2 |
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−7 |
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−17 |
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−11 |
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−27 |
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−7 / 2 |
−17 / 2 |
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−11 |
−27 |
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Add |
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• -11/2 x the 3RD row to the 1st |
• -1 x the 2nd row to the 1st |
• 7/2 x the 3RD row to the 2nd |
• -3 x the 2nd row to the 3rd |
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35 / 2 |
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Multiply the 2 |
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11/ 2 |
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35 / 2 |
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−17 / 2 |
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−7 / 2 |
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−17 / 2 |
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row by -2 |
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0 0 |
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−3 / 2 |
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Solution: x = 1, y = 2, z = 3.
8
Row-Echelon Form of a Matrix
DEFINITION A row of a matrix has k leading zeros if the first k elements of the row are zeros and the (k + 1)th element of the row is not zero.
EXAMPLE
( 0 0 0 0 4 3 0 -1 2)
four leading zeros
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DEFINITION A matrix is in row-echelon form
(ступенчатый вид) if each row has more leading zeros than the row preceding it.
DEFINITION The first nonzero entry in each row of a matrix in row-echelon form is called a pivot
(разрешающий элемент). |
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pivots |
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−0.4 |
−0.3 |
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EXAMPLE |
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0.8 |
−0.2 |
100 |
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0.7 |
210 |
10