Gauss Jordan Method Inverse Of Matrix - WordPress
Gauss Jordan Method&Inverse of Matrix
3 Variables SLEs in Matrix Forma11x a12y a13z b1a21x a22y a23z b2a31x a32y a33z b3Gauss Jordan Method,Augmented Matrix,π11 π12 π13 π₯π1π21 π22 π23 π¦ π2π31 π32 π33 π§π3AR1π11 π12 π13 π1R2π΄/π΅ π21 π22 π23 π2R3π31 π32 π33 π31π12β² π13β² π1β²R1R1/a11, π΄/π΅ π21 π22π23 π2π31 π32 π33 π3X B
R2R2 β a21R1,π΄/π΅ R3R3 β a31R1,π΄/π΅ R2R2/a22β,π΄/π΅ R3R3 β a32βR2,π΄/π΅ 1π12β² π13β² π1β²0π22β² π23β² π2β²π31 π32 π33 π31 π12β² π13β² π1β²0 π22β² π23β² π2β²0 π32β² π33β² π3β²1 π12β² π13β² π1β²01π23β²β² π2β²β²0 π32β² π33β² π3β²1 π12β² π13β² π1β²01π23β²β² π2β²β²00π33β²β² π3β²β²
Case 1 a33ββ 0 Unique SolutionR1R3R1 β a12βR2,R3/a33ββ,R1R2R1 β a13ββR3,R2 β a13ββR3,1000 0 π₯π1β²β²β²1 0 π¦ π2β²β²β²0 1 π§π3β²β²β²1 0 π13β²β² π1β²β²π΄/π΅ 0 1 π23β²β² π2β²β²0 0π3β²β²β²11 0 0 π1β²β²β²π΄/π΅ 0 1 0 π2β²β²β²0 0 1 π3β²β²β²x b1βββy b2βββz b3βββCase 2 a33ββ 0 b3ββ 0 r(A) 2 & r(A/B) 3 r(A) r(A/B) No SolutionCase 3 a33ββ 0 b3ββ 0 r(A) r(A/B) 2 Infinitely Many Solutions
Example 1 Test the Consistency and Solve the following SLEsusing Gauss Jordan Method if possible:x y z 6x - y 2z 53x y z 81 1 1 π₯6In Matrix form SLEs is,AX B1 1 2 π¦ 53 1 1 π§81 1 1 6Augmented Matrix, π΄/π΅ 1 1 2 53 1 1 8
R2R3R3R21 1 1 6R2 β R1, π΄/π΅ 0 2 1 13 1 1 81 161R3 β 3R1, π΄/π΅ 0 2 110 2 2 101 116R3 β R2,π΄/π΅ 0 2 1 10 0 3 9161 1R2/(-2),π΄/π΅ 0 1 1/2 1/20 0 3 9
r(A) r(A/B) 3 n, Therefore, System is Consistent & It has Unique Solution.R1R3R1 β R2,R3/-3,R1R2R1 β 3/2 R3,R2 Β½ R3,x 11 00 10 01003/2 π₯11/2 1/2 π¦ 1/2π§130 0 π₯11 0 π¦ 20 1 π§3y 2z 3Therefore, Solution of the system is unique and it is (x, y , z) (1, 2, 3).
Example 2 Test the Consistency and Solve the following SLEsusing Gauss Jordan Method if possible:3x 2y - 5z 4x y - 2z 15x 3y - 8z 643 2 5 π₯In Matrix form SLEs is,AX B1 1 2 π¦ 165 3 8 π§3 2 5 4Augmented Matrix, π΄/π΅ 1 1 2 15 3 8 6
R1R2R3R21 1 2 1R2,π΄/π΅ 3 2 5 45 3 8 61 1 2R2 β 3R1,π΄/π΅ 0 1 15 3 81 1 2R3 β 5R1,π΄/π΅ 0 1 10 2 21 1 2R2/(-1),π΄/π΅ 0 1 10 2 21161111 11
R31 1 2 1π΄/π΅ 0 1 1 10 0 0 1R3 2R2,r(A) 2&r(A/B) 3 r(A) r(A/B)Therefore, System is inconsistent & It has no solution.1001 2 π₯11 1 π¦ 10 0 π§ 1by row (3), 0 z -10 -1 which Is not possibleTherefor, Solution of given SLE is not possible.
Example 3 Test the Consistency and Solve the following SLEsusing Gauss Jordan Method if possible:2x 2y 2z 0-2x 5y 2z 18x y 4z -12 2 2 π₯0In Matrix form SLEs is, 2 5 2 π¦ 1AX B8 1 4 π§ 12 2 2 0Augmented Matrix, π΄/π΅ 2 5 218 1 4 1
R1R2R3R31R1/2,π΄/π΅ 281R2 2R1, π΄/π΅ 081R3 β 8R1, π΄/π΅ 00R3 R2,1 1 05 2 11 4 11 1 07 4 11 4 1110741 7 4 11 1 1 0π΄/π΅ 0 7 4 10 0 0 0
R2101 1π΄/π΅ 0 1 4/7 1/70 000R2/7,r(A) r(A/B) 2 n,Therefore, System is Consistent & It has Infinitely Many Solutions.R1R1 β R2,z k, k Ρ Ry 4/7 z 1/71 0 3/7 1/7π΄/π΅ 0 1 4/7 1/70 000ππy (1/7) - (4/7) (z) (1/7) - (4/7) (k) x 3/7 z -1/7 (x, y , z) π{( πx πππ,πππ π ππ,ππππk) / k Ρ R}.ππ,πkΡR
Do you think there is a need of newmethod for solving Inverse of a Matrix?1. Yes2. NoVOTE INDIVIDUALLY IN CHAT BOX [30 sec]
Would you like to use Adjoint Method tofind inverse of 4th Order and higher orderMatrix?1. Yes2. NoVOTE INDIVIDUALLY IN CHAT BOX [30 sec]
Inverse of a Matrix by Gauss Jordan MethodThe inverse of an n n matrix A is an n n matrix B having theproperty thatAB BA I[A / I]RREF[I / A-1 ]B is called the inverse of A and is usually denoted by A-1 .If a square matrix has no zero rows in its Row Echelon form orReduced Row Echelon form then inverse of Matrix exists and itis said to be invertible or nonsingular Matrix.If Row Echelon form and Reduced Row Echelon form of Matrixpossess zero row then inverse of Matrix does not exist and theMatrix is said to be singular.One can solve System of Linear Equations AX B if Inverse ofA exists, which yields X A-1 B.
1Example 1 Find the Inverse of Matrix π΄ 13Gauss Jordan Method if possible.1 1 1 1 0Let Augmented Matrix be, [π΄/πΌ] 1 1 2 0 13 1 1 0 01 111R2R2 β R1,[π΄/πΌ] 0 2 1 1R3R3 β 3R1,0 2 2 3R3R2R3 β R2,R2/(-2),1 1[π΄/πΌ] 0 10 01 1/2 31 1 1 21 1using0010 01 00 11001/2 1/2 0 2 11As given square matrix A has no zero rows in its Row Echelon form orReduced Row Echelon form the inverse of Matrix exists.
1[π΄/πΌ] 000 3/2 1/21 1/2 1/2012/3R1R3R1 β R2,R3/-3,R1R21 0R1 β 3/2 R3,[π΄/πΌ] 0 1R2 Β½ R3,0 0 π΄ 11 25 6231/2 1/21/300 1/30 1/200 5/6 1/31 2/31/31/2 1/6 1/310211 3611 33 [I / A-1 ]
Example 2 Solve the following SLEs using Matrix InversionMethod if possible: x y z 61 1 1 π₯6x - y 2z 5 OR 1 1 2 π¦ 53x y z 83 1 1 π§8AX B1 1 1-1 BX Aπ΄ 1 1 211 03 1 1 π΄ 111 022511 636211 333 π 221511 65 2 636 83211 333 x 1y 2z 3
2Example 3 Find the Inverse of Matrix π΄ 28Gauss Jordan Method if possible.2 2 2[π΄/πΌ] 2 5 28 1 4R2R3R3R22 25 21 41 00 10 0R2 2R1,R3 β 8R1,01 1 10 R1R1/2, [π΄/πΌ] 2 5 218 1 41 11 1/2 0 0[π΄/πΌ] 0 7411 00 7 4 4 0 1R3 R2,R2/7,1[π΄/πΌ] 00111/21 4/7 1/700 301/71using1/200010001As given square matrix A has zero row in its Row Echelon form or ReducedRow Echelon form the inverse of Matrix does not exist.001
1Example 4 Find the Inverse of Matrix π΄ 22Method if possible.112Let Augmented Matrix be, [π΄/πΌ] 21R2R3R4R2332534151441100033250100R2 β 2R1,R3 β 2R1,R4 β R1,10[π΄/πΌ] 003 3 423 2 5212201 2 2 1R2/ β3,10[π΄/πΌ] 0031 4232/3 521 2/32034150010010012/3 2 114using Gauss Jordan410001001000010 1/30000100001
R1R3R4R3R4R1 β 3R2,R3 4R2,R4 β 2R2,3R3/β7,3R4,R1R2R4R1 β R3,R2 - 2R3/3,R4 β 2R3,R47R4/24,10[π΄/πΌ] 00010012/3 7/32/310[π΄/πΌ] 00010012/31210[π΄/πΌ] 000100001010[π΄/πΌ] 00010000103 2/3 2/34/33 2/32/74 12/32/3 7/31 1/3 4/32/300100001 12/3 2/7 71 1/34/7200 3/70000319/7 6/72/724/75/76/7 2/7 45/73/7 5/74/76/73/72/7 3/76/7000319/7 6/72/715/76/7 2/7 15/83/7 5/74/72/83/72/7 3/72/80007/8As given square matrix A has no zero rows in its Row Echelon form or Reduced RowEchelon form the inverse of Matrix exists.
R1R2R3R1 β 19R4/7,R2 6R4/7,R3 β 2R4/7,10[π΄/πΌ] 00010035286 π΄ 1 28815 8 28484 8400100001 35/8 6/82/8 15/8 2/8 4/84/82/8 2/84/8 4/82/8 19/86/8 2/87/819868284 8282 87888351 6 8 2 15 2 442 24 42 196 27 [I / A-1 ]
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Inverse of a Matrix by Gauss Jordan Method The inverse of an n n matrix A is an n n matrix B having the property that AB BA I [A / I] [I / A-1] B is called the inverse of A and is usually denoted by A-1. If a square matrix has no zero rows in its Row Echelon form or Reduced Row Echelon fo
EliminaciΓ³n Gaussiana, eliminaciΓ³n de Gauss o eliminaciΓ³n de Gauss-Jordan: En . matemΓ‘ticas, la eliminaciΓ³n gaussiana, eliminaciΓ³n de Gauss o eliminaciΓ³n de Gauss - Jordan, llamadas asΓ debido a Carl Friedrich Gauss y Wilhelm Jordan , son algoritmos del Γ‘lgebra lineal para determinar las soluciones de un sistema de ecuaciones lineales y
B.Inverse S-BOX The inverse S-Box can be defined as the inverse of the S-Box, the calculation of inverse S-Box will take place by the calculation of inverse affine transformation of an Input value that which is followed by multiplicative inverse The representation of Rijndael's inverse S-box is as follows Table 2: Inverse Sbox IV.
Cryptomania. Gauss vs. BΓΌttner BΓΌttnerβsgoal: embarrass Gauss Come up with a problem which Gauss finds difficult but BΓΌttnercan solve quickly 1.Come up with a graph and a Vertex Cover together 2.Give the graph to Gauss 3.When Gauss i
Inverse functions mc-TY-inverse-2009-1 An inverse function is a second function which undoes the work of the ο¬rst one. In this unit we describe two methods for ο¬nding inverse functions, and we also explain that the domain of a function may need to be restricted before an inverse function can exist.
Feb 02, 2020Β Β· The GaussβJordan Method The GaussβJordan elimination method is a technique for solving systems of linear equations of any size. This method involves a sequence of operations on a system of linear equations to obtain at each stage an equivalent systemβthat is, a system having the same s
En la realizaciΓ³n del canon de gauss se aplica las siguientes leyes : Ley de Gauss: El flujo del campo elΓ©ctrico a travΓ©s de cualquier superficie cerrada es igual a la carga q contenida dentro de la superficie, dividida por la constante Ξ΅0. Fig1. EcuaciΓ³n de ley de Gauss La ley de inducciΓ³n electromagnΓ©tica de Faraday
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Gauge Field Theory Dr. Ben Gripaios CavendishLaboratory, JJThomsonAvenue, Cambridge,CB30HE,UnitedKingdom. January4,2016 E-mail: gripaios@hep.phy.cam.ac.uk. Contents 1 Avantpropos1 2 BedtimeReading2 3 Notationandconventions3 4 Relativisticquantummechanics5 4.1 WhyQMdoesanddoesnβtwork5 4.2 TheKlein-Gordonequation7 4.3 TheDiracequation7 4.4 Maxwellβsequations10 4.5 .
