대수 예제
[122220032]⎡⎢⎣122220032⎤⎥⎦
단계 1
단계 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 column 11 by its cofactor and add.
단계 1.1.1
Consider the corresponding sign chart.
|+-+-+-+-+|∣∣
∣∣+−+−+−+−+∣∣
∣∣
단계 1.1.2
The cofactor is the minor with the sign changed if the indices match a -− position on the sign chart.
단계 1.1.3
The minor for a11a11 is the determinant with row 11 and column 11 deleted.
|2032|∣∣∣2032∣∣∣
단계 1.1.4
Multiply element a11a11 by its cofactor.
1|2032|1∣∣∣2032∣∣∣
단계 1.1.5
The minor for a21a21 is the determinant with row 22 and column 11 deleted.
|2232|∣∣∣2232∣∣∣
단계 1.1.6
Multiply element a21a21 by its cofactor.
-2|2232|−2∣∣∣2232∣∣∣
단계 1.1.7
The minor for a31a31 is the determinant with row 33 and column 11 deleted.
|2220|∣∣∣2220∣∣∣
단계 1.1.8
Multiply element a31a31 by its cofactor.
0|2220|0∣∣∣2220∣∣∣
단계 1.1.9
Add the terms together.
1|2032|-2|2232|+0|2220|1∣∣∣2032∣∣∣−2∣∣∣2232∣∣∣+0∣∣∣2220∣∣∣
1|2032|-2|2232|+0|2220|1∣∣∣2032∣∣∣−2∣∣∣2232∣∣∣+0∣∣∣2220∣∣∣
단계 1.2
00에 |2220|∣∣∣2220∣∣∣을 곱합니다.
1|2032|-2|2232|+01∣∣∣2032∣∣∣−2∣∣∣2232∣∣∣+0
단계 1.3
|2032|∣∣∣2032∣∣∣의 값을 구합니다.
단계 1.3.1
2×22×2 행렬의 행렬식은 |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb 공식을 이용해 계산합니다.
1(2⋅2-3⋅0)-2|2232|+01(2⋅2−3⋅0)−2∣∣∣2232∣∣∣+0
단계 1.3.2
행렬식을 간단히 합니다.
단계 1.3.2.1
각 항을 간단히 합니다.
단계 1.3.2.1.1
22에 22을 곱합니다.
1(4-3⋅0)-2|2232|+01(4−3⋅0)−2∣∣∣2232∣∣∣+0
단계 1.3.2.1.2
-3−3에 00을 곱합니다.
1(4+0)-2|2232|+01(4+0)−2∣∣∣2232∣∣∣+0
1(4+0)-2|2232|+01(4+0)−2∣∣∣2232∣∣∣+0
단계 1.3.2.2
44를 00에 더합니다.
1⋅4-2|2232|+01⋅4−2∣∣∣2232∣∣∣+0
1⋅4-2|2232|+01⋅4−2∣∣∣2232∣∣∣+0
1⋅4-2|2232|+01⋅4−2∣∣∣2232∣∣∣+0
단계 1.4
|2232|∣∣∣2232∣∣∣의 값을 구합니다.
단계 1.4.1
2×22×2 행렬의 행렬식은 |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb 공식을 이용해 계산합니다.
1⋅4-2(2⋅2-3⋅2)+01⋅4−2(2⋅2−3⋅2)+0
단계 1.4.2
행렬식을 간단히 합니다.
단계 1.4.2.1
각 항을 간단히 합니다.
단계 1.4.2.1.1
22에 22을 곱합니다.
1⋅4-2(4-3⋅2)+01⋅4−2(4−3⋅2)+0
단계 1.4.2.1.2
-3−3에 22을 곱합니다.
1⋅4-2(4-6)+01⋅4−2(4−6)+0
1⋅4-2(4-6)+01⋅4−2(4−6)+0
단계 1.4.2.2
44에서 66을 뺍니다.
1⋅4-2⋅-2+01⋅4−2⋅−2+0
1⋅4-2⋅-2+01⋅4−2⋅−2+0
1⋅4-2⋅-2+01⋅4−2⋅−2+0
단계 1.5
행렬식을 간단히 합니다.
단계 1.5.1
각 항을 간단히 합니다.
단계 1.5.1.1
44에 11을 곱합니다.
4-2⋅-2+04−2⋅−2+0
단계 1.5.1.2
-2−2에 -2−2을 곱합니다.
4+4+04+4+0
4+4+04+4+0
단계 1.5.2
44를 44에 더합니다.
8+08+0
단계 1.5.3
88를 00에 더합니다.
88
88
88
단계 2
Since the determinant is non-zero, the inverse exists.
단계 3
Set up a 3×63×6 matrix where the left half is the original matrix and the right half is its identity matrix.
[122100220010032001]⎡⎢⎣122100220010032001⎤⎥⎦
단계 4
단계 4.1
Perform the row operation R2=R2-2R1R2=R2−2R1 to make the entry at 2,12,1 a 00.
단계 4.1.1
Perform the row operation R2=R2-2R1R2=R2−2R1 to make the entry at 2,12,1 a 00.
[1221002-2⋅12-2⋅20-2⋅20-2⋅11-2⋅00-2⋅0032001]⎡⎢⎣1221002−2⋅12−2⋅20−2⋅20−2⋅11−2⋅00−2⋅0032001⎤⎥⎦
단계 4.1.2
R2R2을 간단히 합니다.
[1221000-2-4-210032001]⎡⎢⎣1221000−2−4−210032001⎤⎥⎦
[1221000-2-4-210032001]⎡⎢⎣1221000−2−4−210032001⎤⎥⎦
단계 4.2
Multiply each element of R2R2 by -12−12 to make the entry at 2,22,2 a 11.
단계 4.2.1
Multiply each element of R2R2 by -12−12 to make the entry at 2,22,2 a 11.
[122100-12⋅0-12⋅-2-12⋅-4-12⋅-2-12⋅1-12⋅0032001]⎡⎢
⎢⎣122100−12⋅0−12⋅−2−12⋅−4−12⋅−2−12⋅1−12⋅0032001⎤⎥
⎥⎦
단계 4.2.2
R2R2을 간단히 합니다.
[1221000121-120032001]⎡⎢
⎢⎣1221000121−120032001⎤⎥
⎥⎦
[1221000121-120032001]⎡⎢
⎢⎣1221000121−120032001⎤⎥
⎥⎦
단계 4.3
Perform the row operation R3=R3-3R2R3=R3−3R2 to make the entry at 3,23,2 a 00.
단계 4.3.1
Perform the row operation R3=R3-3R2R3=R3−3R2 to make the entry at 3,23,2 a 00.
[1221000121-1200-3⋅03-3⋅12-3⋅20-3⋅10-3(-12)1-3⋅0]⎡⎢
⎢⎣1221000121−1200−3⋅03−3⋅12−3⋅20−3⋅10−3(−12)1−3⋅0⎤⎥
⎥⎦
단계 4.3.2
R3R3을 간단히 합니다.
[1221000121-12000-4-3321]⎡⎢
⎢⎣1221000121−12000−4−3321⎤⎥
⎥⎦
[1221000121-12000-4-3321]⎡⎢
⎢⎣1221000121−12000−4−3321⎤⎥
⎥⎦
단계 4.4
Multiply each element of R3R3 by -14−14 to make the entry at 3,33,3 a 11.
단계 4.4.1
Multiply each element of R3R3 by -14−14 to make the entry at 3,33,3 a 11.
[1221000121-120-14⋅0-14⋅0-14⋅-4-14⋅-3-14⋅32-14⋅1]⎡⎢
⎢⎣1221000121−120−14⋅0−14⋅0−14⋅−4−14⋅−3−14⋅32−14⋅1⎤⎥
⎥⎦
단계 4.4.2
R3R3을 간단히 합니다.
[1221000121-12000134-38-14]⎡⎢
⎢⎣1221000121−12000134−38−14⎤⎥
⎥⎦
[1221000121-12000134-38-14]⎡⎢
⎢⎣1221000121−12000134−38−14⎤⎥
⎥⎦
단계 4.5
Perform the row operation R2=R2-2R3R2=R2−2R3 to make the entry at 2,32,3 a 00.
단계 4.5.1
Perform the row operation R2=R2-2R3R2=R2−2R3 to make the entry at 2,32,3 a 00.
[1221000-2⋅01-2⋅02-2⋅11-2(34)-12-2(-38)0-2(-14)00134-38-14]⎡⎢
⎢
⎢⎣1221000−2⋅01−2⋅02−2⋅11−2(34)−12−2(−38)0−2(−14)00134−38−14⎤⎥
⎥
⎥⎦
단계 4.5.2
R2R2을 간단히 합니다.
[122100010-12141200134-38-14]⎡⎢
⎢⎣122100010−12141200134−38−14⎤⎥
⎥⎦
[122100010-12141200134-38-14]⎡⎢
⎢⎣122100010−12141200134−38−14⎤⎥
⎥⎦
단계 4.6
Perform the row operation R1=R1-2R3R1=R1−2R3 to make the entry at 1,31,3 a 00.
단계 4.6.1
Perform the row operation R1=R1-2R3R1=R1−2R3 to make the entry at 1,31,3 a 00.
[1-2⋅02-2⋅02-2⋅11-2(34)0-2(-38)0-2(-14)010-12141200134-38-14]⎡⎢
⎢
⎢
⎢⎣1−2⋅02−2⋅02−2⋅11−2(34)0−2(−38)0−2(−14)010−12141200134−38−14⎤⎥
⎥
⎥
⎥⎦
단계 4.6.2
R1R1을 간단히 합니다.
[120-123412010-12141200134-38-14]⎡⎢
⎢
⎢⎣120−123412010−12141200134−38−14⎤⎥
⎥
⎥⎦
[120-123412010-12141200134-38-14]⎡⎢
⎢
⎢⎣120−123412010−12141200134−38−14⎤⎥
⎥
⎥⎦
단계 4.7
Perform the row operation R1=R1-2R2R1=R1−2R2 to make the entry at 1,21,2 a 00.
단계 4.7.1
Perform the row operation R1=R1-2R2R1=R1−2R2 to make the entry at 1,21,2 a 00.
[1-2⋅02-2⋅10-2⋅0-12-2(-12)34-2(14)12-2(12)010-12141200134-38-14]⎡⎢
⎢
⎢⎣1−2⋅02−2⋅10−2⋅0−12−2(−12)34−2(14)12−2(12)010−12141200134−38−14⎤⎥
⎥
⎥⎦
단계 4.7.2
R1R1을 간단히 합니다.
[1001214-12010-12141200134-38-14]⎡⎢
⎢
⎢⎣1001214−12010−12141200134−38−14⎤⎥
⎥
⎥⎦
[1001214-12010-12141200134-38-14]⎡⎢
⎢
⎢⎣1001214−12010−12141200134−38−14⎤⎥
⎥
⎥⎦
[1001214-12010-12141200134-38-14]⎡⎢
⎢
⎢⎣1001214−12010−12141200134−38−14⎤⎥
⎥
⎥⎦
단계 5
The right half of the reduced row echelon form is the inverse.
[1214-12-12141234-38-14]⎡⎢
⎢
⎢⎣1214−12−12141234−38−14⎤⎥
⎥
⎥⎦