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Finite Mathematik Beispiele
[332211122]⎡⎢⎣332211122⎤⎥⎦
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.
|1122|∣∣∣1122∣∣∣
Schritt 1.1.4
Multiply element a11a11 by its cofactor.
3|1122|3∣∣∣1122∣∣∣
Schritt 1.1.5
The minor for a12a12 is the determinant with row 11 and column 22 deleted.
|2112|∣∣∣2112∣∣∣
Schritt 1.1.6
Multiply element a12a12 by its cofactor.
-3|2112|−3∣∣∣2112∣∣∣
Schritt 1.1.7
The minor for a13a13 is the determinant with row 11 and column 33 deleted.
|2112|∣∣∣2112∣∣∣
Schritt 1.1.8
Multiply element a13a13 by its cofactor.
2|2112|2∣∣∣2112∣∣∣
Schritt 1.1.9
Add the terms together.
3|1122|-3|2112|+2|2112|3∣∣∣1122∣∣∣−3∣∣∣2112∣∣∣+2∣∣∣2112∣∣∣
3|1122|-3|2112|+2|2112|
Schritt 1.2
Berechne |1122|.
Schritt 1.2.1
Die Determinante einer 2×2-Matrix kann mithilfe der Formel |abcd|=ad-cb bestimmt werden.
3(1⋅2-2⋅1)-3|2112|+2|2112|
Schritt 1.2.2
Vereinfache die Determinante.
Schritt 1.2.2.1
Vereinfache jeden Term.
Schritt 1.2.2.1.1
Mutltipliziere 2 mit 1.
3(2-2⋅1)-3|2112|+2|2112|
Schritt 1.2.2.1.2
Mutltipliziere -2 mit 1.
3(2-2)-3|2112|+2|2112|
3(2-2)-3|2112|+2|2112|
Schritt 1.2.2.2
Subtrahiere 2 von 2.
3⋅0-3|2112|+2|2112|
3⋅0-3|2112|+2|2112|
3⋅0-3|2112|+2|2112|
Schritt 1.3
Berechne |2112|.
Schritt 1.3.1
Die Determinante einer 2×2-Matrix kann mithilfe der Formel |abcd|=ad-cb bestimmt werden.
3⋅0-3(2⋅2-1⋅1)+2|2112|
Schritt 1.3.2
Vereinfache die Determinante.
Schritt 1.3.2.1
Vereinfache jeden Term.
Schritt 1.3.2.1.1
Mutltipliziere 2 mit 2.
3⋅0-3(4-1⋅1)+2|2112|
Schritt 1.3.2.1.2
Mutltipliziere -1 mit 1.
3⋅0-3(4-1)+2|2112|
3⋅0-3(4-1)+2|2112|
Schritt 1.3.2.2
Subtrahiere 1 von 4.
3⋅0-3⋅3+2|2112|
3⋅0-3⋅3+2|2112|
3⋅0-3⋅3+2|2112|
Schritt 1.4
Berechne |2112|.
Schritt 1.4.1
Die Determinante einer 2×2-Matrix kann mithilfe der Formel |abcd|=ad-cb bestimmt werden.
3⋅0-3⋅3+2(2⋅2-1⋅1)
Schritt 1.4.2
Vereinfache die Determinante.
Schritt 1.4.2.1
Vereinfache jeden Term.
Schritt 1.4.2.1.1
Mutltipliziere 2 mit 2.
3⋅0-3⋅3+2(4-1⋅1)
Schritt 1.4.2.1.2
Mutltipliziere -1 mit 1.
3⋅0-3⋅3+2(4-1)
3⋅0-3⋅3+2(4-1)
Schritt 1.4.2.2
Subtrahiere 1 von 4.
3⋅0-3⋅3+2⋅3
3⋅0-3⋅3+2⋅3
3⋅0-3⋅3+2⋅3
Schritt 1.5
Vereinfache die Determinante.
Schritt 1.5.1
Vereinfache jeden Term.
Schritt 1.5.1.1
Mutltipliziere 3 mit 0.
0-3⋅3+2⋅3
Schritt 1.5.1.2
Mutltipliziere -3 mit 3.
0-9+2⋅3
Schritt 1.5.1.3
Mutltipliziere 2 mit 3.
0-9+6
0-9+6
Schritt 1.5.2
Subtrahiere 9 von 0.
-9+6
Schritt 1.5.3
Addiere -9 und 6.
-3
-3
-3
Schritt 2
Since the determinant is non-zero, the inverse exists.
Schritt 3
Set up a 3×6 matrix where the left half is the original matrix and the right half is its identity matrix.
[332100211010122001]
Schritt 4
Schritt 4.1
Multiply each element of R1 by 13 to make the entry at 1,1 a 1.
Schritt 4.1.1
Multiply each element of R1 by 13 to make the entry at 1,1 a 1.
[333323130303211010122001]
Schritt 4.1.2
Vereinfache R1.
[11231300211010122001]
[11231300211010122001]
Schritt 4.2
Perform the row operation R2=R2-2R1 to make the entry at 2,1 a 0.
Schritt 4.2.1
Perform the row operation R2=R2-2R1 to make the entry at 2,1 a 0.
[112313002-2⋅11-2⋅11-2(23)0-2(13)1-2⋅00-2⋅0122001]
Schritt 4.2.2
Vereinfache R2.
[112313000-1-13-2310122001]
[112313000-1-13-2310122001]
Schritt 4.3
Perform the row operation R3=R3-R1 to make the entry at 3,1 a 0.
Schritt 4.3.1
Perform the row operation R3=R3-R1 to make the entry at 3,1 a 0.
[112313000-1-13-23101-12-12-230-130-01-0]
Schritt 4.3.2
Vereinfache R3.
[112313000-1-13-23100143-1301]
[112313000-1-13-23100143-1301]
Schritt 4.4
Multiply each element of R2 by -1 to make the entry at 2,2 a 1.
Schritt 4.4.1
Multiply each element of R2 by -1 to make the entry at 2,2 a 1.
[11231300-0--1--13--23-1⋅1-00143-1301]
Schritt 4.4.2
Vereinfache R2.
[11231300011323-100143-1301]
[11231300011323-100143-1301]
Schritt 4.5
Perform the row operation R3=R3-R2 to make the entry at 3,2 a 0.
Schritt 4.5.1
Perform the row operation R3=R3-R2 to make the entry at 3,2 a 0.
[11231300011323-100-01-143-13-13-230+11-0]
Schritt 4.5.2
Vereinfache R3.
[11231300011323-10001-111]
[11231300011323-10001-111]
Schritt 4.6
Perform the row operation R2=R2-13R3 to make the entry at 2,3 a 0.
Schritt 4.6.1
Perform the row operation R2=R2-13R3 to make the entry at 2,3 a 0.
[112313000-13⋅01-13⋅013-13⋅123-13⋅-1-1-13⋅10-13⋅1001-111]
Schritt 4.6.2
Vereinfache R2.
[112313000101-43-13001-111]
[112313000101-43-13001-111]
Schritt 4.7
Perform the row operation R1=R1-23R3 to make the entry at 1,3 a 0.
Schritt 4.7.1
Perform the row operation R1=R1-23R3 to make the entry at 1,3 a 0.
[1-23⋅01-23⋅023-23⋅113-23⋅-10-23⋅10-23⋅10101-43-13001-111]
Schritt 4.7.2
Vereinfache R1.
[1101-23-230101-43-13001-111]
[1101-23-230101-43-13001-111]
Schritt 4.8
Perform the row operation R1=R1-R2 to make the entry at 1,2 a 0.
Schritt 4.8.1
Perform the row operation R1=R1-R2 to make the entry at 1,2 a 0.
[1-01-10-01-1-23+43-23+130101-43-13001-111]
Schritt 4.8.2
Vereinfache R1.
[100023-130101-43-13001-111]
[100023-130101-43-13001-111]
[100023-130101-43-13001-111]
Schritt 5
The right half of the reduced row echelon form is the inverse.
[023-131-43-13-111]