एलजेब्रा उदाहरण
[022201110]⎡⎢⎣022201110⎤⎥⎦
चरण 1
चरण 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.
चरण 1.1.1
Consider the corresponding sign chart.
|+-+-+-+-+|∣∣
∣∣+−+−+−+−+∣∣
∣∣
चरण 1.1.2
The cofactor is the minor with the sign changed if the indices match a -− position on the sign chart.
चरण 1.1.3
The minor for a11a11 is the determinant with row 11 and column 11 deleted.
|0110|∣∣∣0110∣∣∣
चरण 1.1.4
Multiply element a11a11 by its cofactor.
0|0110|0∣∣∣0110∣∣∣
चरण 1.1.5
The minor for a12a12 is the determinant with row 11 and column 22 deleted.
|2110|∣∣∣2110∣∣∣
चरण 1.1.6
Multiply element a12a12 by its cofactor.
-2|2110|−2∣∣∣2110∣∣∣
चरण 1.1.7
The minor for a13a13 is the determinant with row 11 and column 33 deleted.
|2011|∣∣∣2011∣∣∣
चरण 1.1.8
Multiply element a13a13 by its cofactor.
2|2011|2∣∣∣2011∣∣∣
चरण 1.1.9
Add the terms together.
0|0110|-2|2110|+2|2011|0∣∣∣0110∣∣∣−2∣∣∣2110∣∣∣+2∣∣∣2011∣∣∣
0|0110|-2|2110|+2|2011|0∣∣∣0110∣∣∣−2∣∣∣2110∣∣∣+2∣∣∣2011∣∣∣
चरण 1.2
00 को |0110|∣∣∣0110∣∣∣ से गुणा करें.
0-2|2110|+2|2011|0−2∣∣∣2110∣∣∣+2∣∣∣2011∣∣∣
चरण 1.3
|2110|∣∣∣2110∣∣∣ का मान ज्ञात करें.
चरण 1.3.1
2×22×2 मैट्रिक्स का निर्धारक सूत्र |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb का उपयोग करके पता किया जा सकता है.
0-2(2⋅0-1⋅1)+2|2011|0−2(2⋅0−1⋅1)+2∣∣∣2011∣∣∣
चरण 1.3.2
सारणिक को सरल करें.
चरण 1.3.2.1
प्रत्येक पद को सरल करें.
चरण 1.3.2.1.1
22 को 00 से गुणा करें.
0-2(0-1⋅1)+2|2011|0−2(0−1⋅1)+2∣∣∣2011∣∣∣
चरण 1.3.2.1.2
-1−1 को 11 से गुणा करें.
0-2(0-1)+2|2011|0−2(0−1)+2∣∣∣2011∣∣∣
0-2(0-1)+2|2011|0−2(0−1)+2∣∣∣2011∣∣∣
चरण 1.3.2.2
00 में से 11 घटाएं.
0-2⋅-1+2|2011|0−2⋅−1+2∣∣∣2011∣∣∣
0-2⋅-1+2|2011|0−2⋅−1+2∣∣∣2011∣∣∣
0-2⋅-1+2|2011|0−2⋅−1+2∣∣∣2011∣∣∣
चरण 1.4
|2011|∣∣∣2011∣∣∣ का मान ज्ञात करें.
चरण 1.4.1
2×22×2 मैट्रिक्स का निर्धारक सूत्र |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb का उपयोग करके पता किया जा सकता है.
0-2⋅-1+2(2⋅1-1⋅0)0−2⋅−1+2(2⋅1−1⋅0)
चरण 1.4.2
सारणिक को सरल करें.
चरण 1.4.2.1
22 को 11 से गुणा करें.
0-2⋅-1+2(2-1⋅0)0−2⋅−1+2(2−1⋅0)
चरण 1.4.2.2
22 में से 00 घटाएं.
0-2⋅-1+2⋅20−2⋅−1+2⋅2
0-2⋅-1+2⋅20−2⋅−1+2⋅2
0-2⋅-1+2⋅20−2⋅−1+2⋅2
चरण 1.5
सारणिक को सरल करें.
चरण 1.5.1
प्रत्येक पद को सरल करें.
चरण 1.5.1.1
-2−2 को -1−1 से गुणा करें.
0+2+2⋅20+2+2⋅2
चरण 1.5.1.2
22 को 22 से गुणा करें.
0+2+40+2+4
0+2+40+2+4
चरण 1.5.2
00 और 22 जोड़ें.
2+42+4
चरण 1.5.3
22 और 44 जोड़ें.
66
66
66
चरण 2
Since the determinant is non-zero, the inverse exists.
चरण 3
Set up a 3×63×6 matrix where the left half is the original matrix and the right half is its identity matrix.
[022100201010110001]⎡⎢⎣022100201010110001⎤⎥⎦
चरण 4
चरण 4.1
Swap R2R2 with R1R1 to put a nonzero entry at 1,11,1.
[201010022100110001]⎡⎢⎣201010022100110001⎤⎥⎦
चरण 4.2
Multiply each element of R1R1 by 1212 to make the entry at 1,11,1 a 11.
चरण 4.2.1
Multiply each element of R1R1 by 1212 to make the entry at 1,11,1 a 11.
[220212021202022100110001]⎡⎢
⎢⎣220212021202022100110001⎤⎥
⎥⎦
चरण 4.2.2
R1R1 को सरल करें.
[10120120022100110001]⎡⎢
⎢⎣10120120022100110001⎤⎥
⎥⎦
[10120120022100110001]⎡⎢
⎢⎣10120120022100110001⎤⎥
⎥⎦
चरण 4.3
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
चरण 4.3.1
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
[101201200221001-11-00-120-00-121-0]⎡⎢
⎢⎣101201200221001−11−00−120−00−121−0⎤⎥
⎥⎦
चरण 4.3.2
R3R3 को सरल करें.
[1012012002210001-120-121]⎡⎢
⎢⎣1012012002210001−120−121⎤⎥
⎥⎦
[1012012002210001-120-121]⎡⎢
⎢⎣1012012002210001−120−121⎤⎥
⎥⎦
चरण 4.4
Multiply each element of R2R2 by 1212 to make the entry at 2,22,2 a 11.
चरण 4.4.1
Multiply each element of R2R2 by 1212 to make the entry at 2,22,2 a 11.
[1012012002222212020201-120-121]⎡⎢
⎢
⎢⎣1012012002222212020201−120−121⎤⎥
⎥
⎥⎦
चरण 4.4.2
R2R2 को सरल करें.
[10120120011120001-120-121]⎡⎢
⎢
⎢⎣10120120011120001−120−121⎤⎥
⎥
⎥⎦
[10120120011120001-120-121]⎡⎢
⎢
⎢⎣10120120011120001−120−121⎤⎥
⎥
⎥⎦
चरण 4.5
Perform the row operation R3=R3-R2R3=R3−R2 to make the entry at 3,23,2 a 00.
चरण 4.5.1
Perform the row operation R3=R3-R2R3=R3−R2 to make the entry at 3,23,2 a 00.
[1012012001112000-01-1-12-10-12-12-01-0]⎡⎢
⎢
⎢⎣1012012001112000−01−1−12−10−12−12−01−0⎤⎥
⎥
⎥⎦
चरण 4.5.2
R3R3 को सरल करें.
[10120120011120000-32-12-121]⎡⎢
⎢
⎢⎣10120120011120000−32−12−121⎤⎥
⎥
⎥⎦
[10120120011120000-32-12-121]⎡⎢
⎢
⎢⎣10120120011120000−32−12−121⎤⎥
⎥
⎥⎦
चरण 4.6
Multiply each element of R3 by -23 to make the entry at 3,3 a 1.
चरण 4.6.1
Multiply each element of R3 by -23 to make the entry at 3,3 a 1.
[101201200111200-23⋅0-23⋅0-23(-32)-23(-12)-23(-12)-23⋅1]
चरण 4.6.2
R3 को सरल करें.
[1012012001112000011313-23]
[1012012001112000011313-23]
चरण 4.7
Perform the row operation R2=R2-R3 to make the entry at 2,3 a 0.
चरण 4.7.1
Perform the row operation R2=R2-R3 to make the entry at 2,3 a 0.
[101201200-01-01-112-130-130+230011313-23]
चरण 4.7.2
R2 को सरल करें.
[1012012001016-13230011313-23]
[1012012001016-13230011313-23]
चरण 4.8
Perform the row operation R1=R1-12R3 to make the entry at 1,3 a 0.
चरण 4.8.1
Perform the row operation R1=R1-12R3 to make the entry at 1,3 a 0.
[1-12⋅00-12⋅012-12⋅10-12⋅1312-12⋅130-12(-23)01016-13230011313-23]
चरण 4.8.2
R1 को सरल करें.
[100-16131301016-13230011313-23]
[100-16131301016-13230011313-23]
[100-16131301016-13230011313-23]
चरण 5
The right half of the reduced row echelon form is the inverse.
[-16131316-13231313-23]
