Beispiele
[440231123]⎡⎢⎣440231123⎤⎥⎦
Schritt 1
Schritt 1.1
Choose the row or column with the most 00 elements. If there are no 00 elements choose any row or column. Multiply every element in row 11 by its cofactor and add.
Schritt 1.1.1
Consider the corresponding sign chart.
|+-+-+-+-+|∣∣
∣∣+−+−+−+−+∣∣
∣∣
Schritt 1.1.2
The cofactor is the minor with the sign changed if the indices match a -− position on the sign chart.
Schritt 1.1.3
The minor for a11a11 is the determinant with row 11 and column 11 deleted.
|3123|∣∣∣3123∣∣∣
Schritt 1.1.4
Multiply element a11a11 by its cofactor.
4|3123|4∣∣∣3123∣∣∣
Schritt 1.1.5
The minor for a12a12 is the determinant with row 11 and column 22 deleted.
|2113|∣∣∣2113∣∣∣
Schritt 1.1.6
Multiply element a12a12 by its cofactor.
-4|2113|−4∣∣∣2113∣∣∣
Schritt 1.1.7
The minor for a13a13 is the determinant with row 11 and column 33 deleted.
|2312|∣∣∣2312∣∣∣
Schritt 1.1.8
Multiply element a13a13 by its cofactor.
0|2312|0∣∣∣2312∣∣∣
Schritt 1.1.9
Add the terms together.
4|3123|-4|2113|+0|2312|4∣∣∣3123∣∣∣−4∣∣∣2113∣∣∣+0∣∣∣2312∣∣∣
4|3123|-4|2113|+0|2312|4∣∣∣3123∣∣∣−4∣∣∣2113∣∣∣+0∣∣∣2312∣∣∣
Schritt 1.2
Mutltipliziere 00 mit |2312|∣∣∣2312∣∣∣.
4|3123|-4|2113|+04∣∣∣3123∣∣∣−4∣∣∣2113∣∣∣+0
Schritt 1.3
Berechne |3123|∣∣∣3123∣∣∣.
Schritt 1.3.1
Die Determinante einer 2×22×2-Matrix kann mithilfe der Formel |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb bestimmt werden.
4(3⋅3-2⋅1)-4|2113|+04(3⋅3−2⋅1)−4∣∣∣2113∣∣∣+0
Schritt 1.3.2
Vereinfache die Determinante.
Schritt 1.3.2.1
Vereinfache jeden Term.
Schritt 1.3.2.1.1
Mutltipliziere 33 mit 33.
4(9-2⋅1)-4|2113|+04(9−2⋅1)−4∣∣∣2113∣∣∣+0
Schritt 1.3.2.1.2
Mutltipliziere -2−2 mit 11.
4(9-2)-4|2113|+04(9−2)−4∣∣∣2113∣∣∣+0
4(9-2)-4|2113|+04(9−2)−4∣∣∣2113∣∣∣+0
Schritt 1.3.2.2
Subtrahiere 22 von 99.
4⋅7-4|2113|+04⋅7−4∣∣∣2113∣∣∣+0
4⋅7-4|2113|+04⋅7−4∣∣∣2113∣∣∣+0
4⋅7-4|2113|+04⋅7−4∣∣∣2113∣∣∣+0
Schritt 1.4
Berechne |2113|∣∣∣2113∣∣∣.
Schritt 1.4.1
Die Determinante einer 2×22×2-Matrix kann mithilfe der Formel |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb bestimmt werden.
4⋅7-4(2⋅3-1⋅1)+04⋅7−4(2⋅3−1⋅1)+0
Schritt 1.4.2
Vereinfache die Determinante.
Schritt 1.4.2.1
Vereinfache jeden Term.
Schritt 1.4.2.1.1
Mutltipliziere 22 mit 33.
4⋅7-4(6-1⋅1)+04⋅7−4(6−1⋅1)+0
Schritt 1.4.2.1.2
Mutltipliziere -1−1 mit 11.
4⋅7-4(6-1)+04⋅7−4(6−1)+0
4⋅7-4(6-1)+04⋅7−4(6−1)+0
Schritt 1.4.2.2
Subtrahiere 11 von 66.
4⋅7-4⋅5+04⋅7−4⋅5+0
4⋅7-4⋅5+04⋅7−4⋅5+0
4⋅7-4⋅5+04⋅7−4⋅5+0
Schritt 1.5
Vereinfache die Determinante.
Schritt 1.5.1
Vereinfache jeden Term.
Schritt 1.5.1.1
Mutltipliziere 44 mit 77.
28-4⋅5+028−4⋅5+0
Schritt 1.5.1.2
Mutltipliziere -4−4 mit 55.
28-20+028−20+0
28-20+028−20+0
Schritt 1.5.2
Subtrahiere 2020 von 2828.
8+08+0
Schritt 1.5.3
Addiere 88 und 00.
88
88
88
Schritt 2
Since the determinant is non-zero, the inverse exists.
Schritt 3
Set up a 3×63×6 matrix where the left half is the original matrix and the right half is its identity matrix.
[440100231010123001]⎡⎢⎣440100231010123001⎤⎥⎦
Schritt 4
Schritt 4.1
Multiply each element of R1R1 by 1414 to make the entry at 1,11,1 a 11.
Schritt 4.1.1
Multiply each element of R1R1 by 1414 to make the entry at 1,11,1 a 11.
[444404140404231010123001]⎡⎢
⎢⎣444404140404231010123001⎤⎥
⎥⎦
Schritt 4.1.2
Vereinfache R1R1.
[1101400231010123001]⎡⎢
⎢⎣1101400231010123001⎤⎥
⎥⎦
[1101400231010123001]⎡⎢
⎢⎣1101400231010123001⎤⎥
⎥⎦
Schritt 4.2
Perform the row operation R2=R2-2R1R2=R2−2R1 to make the entry at 2,12,1 a 00.
Schritt 4.2.1
Perform the row operation R2=R2-2R1R2=R2−2R1 to make the entry at 2,12,1 a 00.
[11014002-2⋅13-2⋅11-2⋅00-2(14)1-2⋅00-2⋅0123001]⎡⎢
⎢⎣11014002−2⋅13−2⋅11−2⋅00−2(14)1−2⋅00−2⋅0123001⎤⎥
⎥⎦
Schritt 4.2.2
Vereinfache R2R2.
[1101400011-1210123001]⎡⎢
⎢⎣1101400011−1210123001⎤⎥
⎥⎦
[1101400011-1210123001]⎡⎢
⎢⎣1101400011−1210123001⎤⎥
⎥⎦
Schritt 4.3
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
Schritt 4.3.1
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
[1101400011-12101-12-13-00-140-01-0]⎡⎢
⎢
⎢⎣1101400011−12101−12−13−00−140−01−0⎤⎥
⎥
⎥⎦
Schritt 4.3.2
Vereinfache R3R3.
[1101400011-1210013-1401]⎡⎢
⎢
⎢⎣1101400011−1210013−1401⎤⎥
⎥
⎥⎦
[1101400011-1210013-1401]⎡⎢
⎢
⎢⎣1101400011−1210013−1401⎤⎥
⎥
⎥⎦
Schritt 4.4
Perform the row operation R3=R3-R2R3=R3−R2 to make the entry at 3,23,2 a 00.
Schritt 4.4.1
Perform the row operation R3=R3-R2R3=R3−R2 to make the entry at 3,23,2 a 00.
[1101400011-12100-01-13-1-14+120-11-0]⎡⎢
⎢
⎢⎣1101400011−12100−01−13−1−14+120−11−0⎤⎥
⎥
⎥⎦
Schritt 4.4.2
Vereinfache R3R3.
[1101400011-121000214-11]⎡⎢
⎢
⎢⎣1101400011−121000214−11⎤⎥
⎥
⎥⎦
[1101400011-121000214-11]⎡⎢
⎢
⎢⎣1101400011−121000214−11⎤⎥
⎥
⎥⎦
Schritt 4.5
Multiply each element of R3R3 by 1212 to make the entry at 3,33,3 a 11.
Schritt 4.5.1
Multiply each element of R3R3 by 1212 to make the entry at 3,33,3 a 11.
[1101400011-1210020222142-1212]⎡⎢
⎢
⎢
⎢⎣1101400011−1210020222142−1212⎤⎥
⎥
⎥
⎥⎦
Schritt 4.5.2
Vereinfache R3R3.
[1101400011-121000118-1212]⎡⎢
⎢
⎢⎣1101400011−121000118−1212⎤⎥
⎥
⎥⎦
[1101400011-121000118-1212]⎡⎢
⎢
⎢⎣1101400011−121000118−1212⎤⎥
⎥
⎥⎦
Schritt 4.6
Perform the row operation R2=R2-R3R2=R2−R3 to make the entry at 2,32,3 a 00.
Schritt 4.6.1
Perform the row operation R2=R2-R3R2=R2−R3 to make the entry at 2,32,3 a 00.
[11014000-01-01-1-12-181+120-1200118-1212]⎡⎢
⎢
⎢⎣11014000−01−01−1−12−181+120−1200118−1212⎤⎥
⎥
⎥⎦
Schritt 4.6.2
Vereinfache R2R2.
[1101400010-5832-1200118-1212]⎡⎢
⎢
⎢⎣1101400010−5832−1200118−1212⎤⎥
⎥
⎥⎦
[1101400010-5832-1200118-1212]⎡⎢
⎢
⎢⎣1101400010−5832−1200118−1212⎤⎥
⎥
⎥⎦
Schritt 4.7
Perform the row operation R1=R1-R2R1=R1−R2 to make the entry at 1,21,2 a 00.
Schritt 4.7.1
Perform the row operation R1=R1-R2R1=R1−R2 to make the entry at 1,21,2 a 00.
[1-01-10-014+580-320+12010-5832-1200118-1212]⎡⎢
⎢
⎢⎣1−01−10−014+580−320+12010−5832−1200118−1212⎤⎥
⎥
⎥⎦
Schritt 4.7.2
Vereinfache R1R1.
[10078-3212010-5832-1200118-1212]⎡⎢
⎢
⎢⎣10078−3212010−5832−1200118−1212⎤⎥
⎥
⎥⎦
[10078-3212010-5832-1200118-1212]⎡⎢
⎢
⎢⎣10078−3212010−5832−1200118−1212⎤⎥
⎥
⎥⎦
[10078-3212010-5832-1200118-1212]⎡⎢
⎢
⎢⎣10078−3212010−5832−1200118−1212⎤⎥
⎥
⎥⎦
Schritt 5
The right half of the reduced row echelon form is the inverse.
[78-3212-5832-1218-1212]
