A.matrix algebra for structural analysisdoc

Department of Civil Engineering NPIC

A. m:aRTIssRmab;karviPaKeRKOgbgÁúM
(Matrix algebra for structural Analysis)

A >!> niymn½y nigRbePTm:aRTIs (basic definitions and types of matrices)
edaysarPaBcaM)ac;énkMuBüÚT½r dUcenHkarGnuvtþm:aRTIssRmab;karviPaKeRKOgbgÁúMmanlkçN³
TUlMTUlay. m:aRTIspþl;nUvmeFüa)ayd¾smRsbsRmab;karviPaKenH edaysarvamanlkçN³gayRsYl
kñúgkarsresrrUbmnþkñúgTMrg;c,as;las; ehIybnÞab;mkedaHRsaym:aRTIsedayeRbIkMuBüÚT½r. sRmab;mUl
ehtuenH visVkreRKOgbgÁúMRtUvEtyl;BIvaeGay)anc,as;.
m:aRTIs³ m:aRTIsCakartMerobelxkñúgTMrg;ctuekaNEdlamnCYredk m nigCYrQr n . elx ¬EdleKehA
faFatu¦ RtUv)antMerobenAkñúgekñób. ]TahrN_ m:aRTIs A RtUv)ansresrCa³
⎡ a11 a12 L a1n ⎤
⎢a ⎥
A= ⎢ 21 a 22 L a 2 n ⎥
⎢ M ⎥
⎢ ⎥
⎣a m1 a m 2 L a mn ⎦
m:aRTIsEbbenHRtUv)aneKehAfam:aRTIs m × n . cMNaMfasnÞsSn_TImYysRmab;FatunImYy²CaTItaMgCUr
edkrbs;va ehIysnÞsSn_TIBIrCaTItaMgCYrQrrbs;va. CaTUeTA aij CaFatuEdlmanTItaMgenAkñúgCYredk
TI i nigCYrQrTI j .
m:aRTIsCYredk³ RbsinebIm:aRTIspSMeLIgEtBIFatuenAkñúgCYredkeTal eKehAvafaCam:aRTIsCYredk.
]TahrN_ m:aRTIsCYredk1 × n RtUv)aneKsresrCa
A = [a1 a 2 L a n ]
enATIenH eKeRbIEtsnÞsSn_eTaledIm,IsMKal;Fatu edaysareKdwgfasnÞsSn_CYrQresμInwg1 eBalKW
a1 = a11 , a 2 = a12 , nigbnþbnÞab;.

m:aRTIsCYrQr³ m:aRTIsEdlmanFatuKrelIKñakñúgCYreTal eKeGayeQμaHvafam:aRTIsCYrQr.
m:aRTIsCYrQr m ×1 KW
⎡ a1 ⎤
⎢a ⎥
A=⎢ 2⎥
⎢ M ⎥
⎢ ⎥
⎣a m ⎦
enATIenH Fatu a1 = a11 , a 2 = a 21 , nigbnþbnÞab;.

m:aRTIssRmab;karviPaKeRKOgbgÁúM T.Chhay -538

mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa

m:aRTIskaer³ enAeBlcMnYnCYredkrbs;m:aRTIsesμInwgcMnYnCYrQr m:aRTIsenHmaneQμaHfam:aRTIskaer.
m:aRTIskaer n × n manTRmg;
⎡ a11 a12 L a1n ⎤
⎢a ⎥
A= ⎢ 21 a 22 L a 2 n ⎥
⎢ M ⎥
⎢ ⎥
⎣ a n1 a n 2 L a nn ⎦
m:aRTIsGgát;RTUg³ enAeBlRKb;FatuTaMgGs;rbs;m:aRTIsesμIsUnü elIkElgEtFatutamGgát;RTUg m:aRTIs
enHmaneQμaHfam:aRTIsGgát;RTUg. ]TahrN_
⎡a11 0 0 ⎤
⎢ 0 a
A=⎢ 0 ⎥
22 ⎥
⎢ 0
⎣ 0 a 33 ⎥
⎦
m:aRTIsÉktþa³ m:aRTIsÉktþaCam:aRTIsGgát;RTUgEdlFatutamGgát;esμInwgcMnYnÉktþa. ]TahrN_
⎡1 0 0 ⎤
I = ⎢0 1 0 ⎥
⎢ ⎥
⎢0 0 1 ⎥
⎣ ⎦
m:aRTIssIuemRTI³ m:aRTIskaermanlkçN³sIuemRTIluHRtaEt aij = a ji . ]TahrN_
⎡ 3 5 2⎤
A = ⎢5 − 1 4⎥
⎢ ⎥
⎢2 4 8⎥
⎣ ⎦

A >@> RbmaNviFIma:RTIs (matrix operation)
smPaBrbs;m:aRTIs³ m:aRTIs A nig B esμIKñaRbsinebIm:aRTIsTaMgBIrmanlMdab;esμIKña ehIyFatuRtUvKña
rbs;m:aRTIsTaMgBIresμIKña eBalKW aij = bij . ]TahrN_ RbsinebI
⎡2 6 ⎤ ⎡2 6 ⎤
A=⎢ ⎥ B=⎢ ⎥
⎣4 − 3⎦ ⎣4 − 3⎦
enaH A = B
plbUk nigpldkrbs;m:aRTIs³ eKGaceFVIRbmaNviFIbUk b¤dkm:aRTIseTA)anluHRtavaCam:aRTIsEdlman
lMdab;esμIKña. lT§plRtUv)anTTYlBIkarbUk nigdkFatuRtUvKña. ]TahrN_ RbsinebI
⎡6 7 ⎤ ⎡− 5 8 ⎤
A=⎢ ⎥ B=⎢ ⎥
⎣2 − 1⎦ ⎣ 1 4⎦
⎡1 15⎤ ⎡11 − 1⎤
enaH A+ B=⎢ ⎥ A− B = ⎢ ⎥
⎣3 3 ⎦ ⎣ 1 − 5⎦

Matrix algebra for structural analysis T.Chhay -539


RbmaNviFIKuNedaysáaElr³ enAeBlm:aRTIsRtUv)anKuNedaysáaElr FatunImYy²rbs;m:aRTIsRtUv)an
KuNnwgTMhMsáaElrenH. ]TahrN_ RbsinebI
⎡4 1 ⎤
A=⎢ ⎥ k = −6
⎣6 − 2⎦
⎡− 24 − 6⎤
enaH kA = ⎢ ⎥
⎣ − 36 12 ⎦
RbmaNviFIKuNm:aRTIs³ eKGacKuNm:aRTIsBIr A nig B bBa©ÚlKña)anluHRtaEtvaRsbKña. lkçxNÐenH
GacbMeBj)an RbsinebIcMnYnrbs;CYrQrenAkñúgm:aRTIs A esμInwgcMnYnCYredkkñúgm:aRTIs B . ]TahrN_
RbsinebI
⎡a a ⎤ ⎡ b11 b12 b13 ⎤
A = ⎢ 11 12 ⎥ B=⎢ ⎥ (A-1)
⎣a 21 a 22 ⎦ ⎣b21 b22 b23 ⎦
enaHeKGackMNt; AB edaysar A manBIrCYrQr ehIy B manBIrCYredk. b:uEnþcMNaMfa eKminGackMNt;
BA )aneT. ehtuGVI?

RbsinebIeKKuNm:aRTIs A EdlmanlMdab; (m × n) CamYynwgm:aRTIs B EdlmanlMdab; (n × q )
eyIgnwgTTYl)anm:aRTIs C EdlmanlMdab; (m × q ) eBalKW
A B = C
(m × n ) (n × q ) (m × q )
eKrkFaturbs;m:aRTIs C edayeRbIFatu aij rbs;m:aRTIs A nig bij rbs;m:aRTIs B dUcxageRkam
n
cij = ∑ aik bkj (A-2)
k =1

eKGacBnül;viFIsaRsþénrUbmnþenHeday]TahrN_samBaØxøH. eKman
⎡ 2⎤
⎡ 2 4 3⎤
A=⎢ ⎥ B = ⎢6 ⎥
⎢ ⎥
⎣− 1 6 1⎦ ⎢7 ⎥
⎣ ⎦
tamkarGegát eKGaceFVIRbmaNviFIKuN C = AB edaysarm:aRTIsTaMgenHCam:aRTIsRsbKña eBalKW A
manCYrQrbI ehIym:aRTIs B manCYredkbI. tamsmIkar A-1 RbmaNviFIKuNm:aRTIseFVIeGaym:aRTIs C
manCYredkBIr nigCYrQrmYy. lT§plEdlTTYl)andUcxageRkam
c11 : KuNFatuenAkñúgCYredkTImYyrbs; A CamYynwgFatuRtUvKñaenAkñúgCYrQrrbs; B ehIybUklT§pl

TaMgenaHbBa©ÚlKña eBalKW
c11 = c1 = 2(2 ) + 4(6 ) + 3(7 ) = 49



c21 : KuNFatuenAkñúgCYredkTIBIrrbs; A CamYynwgFatuRtUvKñaenAkñúgCYrQrrbs; B ehIybUklT§pl
TaMgenaHbBa©ÚlKña eBalKW
c21 = c2 = −1(2 ) + 6(6 ) + 1(7 ) = 41
⎡49⎤
dUcenH C=⎢ ⎥
⎣ 41⎦
dUc]TahrN_TIBIr eKman
⎡ 5 3⎤
⎡ 2 7⎤
A = ⎢ 4 1⎥
⎢ ⎥ B=⎢ ⎥
⎢ − 2 8⎥ ⎣− 3 4⎦
⎣ ⎦
enATIenH eKGacrkplKuN C = AB edaysar A manCYrQrBIr ehiy B manCYredkBIr. m:aRTIs C nwg
manCYredkbI nigCYrQrBIr. eKGacrkFaturbs;m:aRTIs C dUcxageRkam³
c11 = 5(2 ) + 3(− 3) = 1 ¬CYredkTImYyrbs; A KuNnwgCYrQrTImYyrbs; B ¦
c12 = 5(7 ) + 3(4) = 47 ¬CYredkTImYyrbs; A KuNnwgCYrQrTIBIrrbs; B ¦
c21 = 4(2) + 1(− 3) = 5 ¬CYredkTIBIrrbs; A KuNnwgCYrQrTImYyrbs; B ¦
c22 = 4(7 ) + 1(4 ) = 32 ¬CYredkTIBIrrbs; A KuNnwgCYrQrTIBIrrbs; B ¦
c31 = −2(2) + 8(− 3) = −28 ¬CYredkTIbIrbs; A KuNnwgCYrQrTImYyrbs; B ¦
c32 = −2(7 ) + 8(4) = 18 ¬CYredkTIbIrbs; A KuNnwgCYrQrTIBIrrbs; B ¦
lT§plsRmab;RbmaNviFIKuNGnuvtþtamsmIkar A-2. dUcenH
⎡ 1 47 ⎤
C = ⎢ 5 32 ⎥
⎢ ⎥
⎢− 28 18 ⎥
⎣ ⎦
eKk¾RtUvcMNaMplEdrfa eKminGaceFVIplKuN BA )aneT edaysarkarsresrEbbenHm:aRTIsminRsbKña.
c,ab;xageRkamGnuvtþcMeBaHRbmaNviFIKuN
!> CaTUeTA RbmaNviFIKuNm:aRTIsminmanlkçN³qøas;eT
AB ≠ BA (A-3)
@> eKGacBnøatRbmaNviFIKuNm:aRTIs
A(B + C ) = AB + AC (A-4)
#> eKGacpþúMRbmaNviFIKuNm:aRTIs
A(BC ) = ( AB )C (A-5)
Transposed matrix: eKGaceFVI transposed m:aRTIsedaybþÚrCYredk nigCYrQrrbs;va. ]TahrN_
RbsinebI


⎡ a11 a12 a13 ⎤
A = ⎢a21 a22 a23 ⎥
⎢ ⎥ B = [b1 b2 b3 ]
⎢ a31 a32 a33 ⎥
⎣ ⎦
⎡ a11 a21 a31 ⎤ ⎡ b1 ⎤
enaH A = ⎢ a12 a22 a32 ⎥
T
⎢ ⎥ B = ⎢b2 ⎥
T
⎢ ⎥
⎢ A13 A23 A33 ⎥
⎣ ⎦ ⎢b3 ⎥
⎣ ⎦
cMNaMfa AB minRsbKña dUcenHeKminGaceFVIRbmaNviFIKuNm:aRTIseT. ¬ A manbICYrQr ehIy B man
CYredkmYy¦. mü:agvijeTot eKGaceFVIRbmaNviFIKuN ABT edaysarenATIenHm:aRTIsTaMgBIrRsbKña
¬ A manbICYrQr ehIy BT manbICYredk¦. xageRkamCalkçN³sRmab; transposed matrix
( A + B )T = AT + BT (A-6)
(kA)T = kAT (A-7)
( AB )T = BT AT (A-8)
eKGacbgðajlkçN³cugeRkayeday]TahrN_. RbsinebI
⎡6 2 ⎤ ⎡4 3⎤
A=⎢ ⎥ B=⎢ ⎥
⎣1 − 3⎦ ⎣2 5⎦
bnÞab;mk tamsmIkar A-8
T
⎛ ⎡6 2 ⎤ ⎡4 3⎤ ⎞ ⎡ 4 2⎤ ⎡6 1 ⎤
⎜⎢ ⎟
⎜ 1 − 2⎥ ⎢2 5⎥ ⎟ = ⎢3 5⎥ ⎢2 − 3⎥
⎝⎣ ⎦⎣ ⎦⎠ ⎣ ⎦⎣ ⎦
T
⎛ ⎡ 28 28 ⎤ ⎞ ⎡28 − 2 ⎤
⎜⎢ ⎥ ⎟ = ⎢28 − 12⎥
⎜ − 2 − 12 ⎟
⎝⎣ ⎦⎠ ⎣ ⎦
⎡28 − 2 ⎤ ⎡28 − 2 ⎤
⎢28 − 12⎥ = ⎢28 − 12⎥
⎣ ⎦ ⎣ ⎦
karEbgEckm:aRTIsCaRkum³ eKGacEbgEckm:aRTIsCam:aRTIsrgedaykarEbgEckvaCaRkum. ]TahrN_
⎡ a11 a12 a13 a14 ⎤
⎡A A ⎤
A = ⎢a21 a22 a23 a24 ⎥ = ⎢ 11 12 ⎥
⎢ ⎥
⎢ a31 a32 a33 a34 ⎥ ⎣ 21 22 ⎦
A A
⎣ ⎦
enATIenHm:aRTIsrgKW
A11 = [a11 ] A12 = [a12 a13 a14 ]
⎡a ⎤ ⎡a a a ⎤
A21 = ⎢ 21 ⎥ A22 = ⎢ 22 23 24 ⎥
⎣ a31 ⎦ ⎣ a32 a33 a34 ⎦
c,ab;rbs;m:aRTIsGnuvtþeTAelIm:aRTIsEdlEbgEckCaRkumluHRtaEtkarEbgEckmanlkçN³RsbKña.
]TahrN_ eKGacbUk nigdkm:aRTIsrg A nig B luHRtaEtvamancMnYnCYredk nigCYrQresμIKña. dUcKña eK
GaceFVIRbmaNviFIKuNm:aRTIsluHRtaEtcMnYnCYredk nigcMnYnCYrQrRtUvKñaénm:aRTIs A nigm:aRTIs B esμIKña


ehIym:aRTIsrgrbs;vaesμIKña. ]TahrN_ RbsinebI
⎡ 4 1 − 1⎤ ⎡ 2 − 1⎤
A = ⎢ − 2 0 − 5⎥
⎢ ⎥ B = ⎢ 0 − 8⎥
⎢ ⎥
⎢6 3 8⎥
⎣ ⎦ ⎢7 4 ⎥
⎣ ⎦
bnÞab;mk eKGaceFVIplKuN AB edaysarcMnUnCYrQrrbs; A esμInwgcMnYnCYredkrbs; B . dUcKña
m:aRTIsEdlbMEbkCaRkummanlkçN³RsbKñasRmab;RbmaNviFIKuN edaysar A RtUv)anbMEbkCa
CYrQrBIr ehIy B RtUv)anbMEbkCaCYredkBIr eBalKW
⎡ A A ⎤⎡ B ⎤ ⎡ A B + A B ⎤
AB = ⎢ 11 12 ⎥ ⎢ 11 ⎥ = ⎢ 11 11 12 21 ⎥
⎣ A21 A22 ⎦ ⎣ B21 ⎦ ⎣ A21B11 + A22 B21 ⎦
plKuNrbs;m:aRTIsrgKW
⎡ 4 1⎤ ⎡2 − 1⎤ ⎡ 8 4⎤
A11B11 = ⎢ ⎥⎢ ⎥=⎢ ⎥
⎣ − 2 0 ⎦ ⎣0 8 ⎦ ⎣ − 4 2 ⎦
⎡ − 1⎤ ⎡ −7 −4 ⎤
A12 B21 = ⎢ ⎥[7 4] = ⎢ ⎥
⎣− 5⎦ ⎣− 35 − 20⎦
⎡2 − 1⎤
A21B11 = [6 3]⎢ ⎥ = [12 18]
⎣0 8 ⎦
A22 B21 = [8][7 4] = [56 32]
⎡ ⎡ 8 4⎤ ⎡ − 7 − 4 ⎤ ⎤ ⎡ 1 0 ⎤
dUcenH ⎢⎢ ⎥ + ⎢− 35 − 20⎥ ⎥ = ⎢− 39 − 18⎥
AB = ⎣− 4 2⎦ ⎣
⎢ ⎦⎥ ⎢ ⎥
⎢
⎣ [12 18] + [56 32] ⎥ ⎢ 68 50 ⎥
⎦ ⎣ ⎦

A >#> edETmINg; (Determinants)
enAkñúgkfaxNÐbnÞab; eyIgnwgerobrab;BIrebobcRmas;m:aRTIs. edaysarRbmaNviFIenHRtUvkar
karKNnaedETmINg;rbs;m:aRTIs eyIgnwgerobrab;BIrlkçN³mUldæanrbs;edETmINg;.
edETmINg;CakartMerobelxCaTRmg;kaeredaysßitenAkñúgr)arbBaÄr. ]TahrN_ edETmINg;
lMdab; n ¬EdlmanCYredk n nigCYrQr n ¦ KW
a11 a12 L a1n
a21 a22 L a2 n
A= (A-9)
M
an1 an 2 L ann

karKNnaedETmINg;enHeFVIeGayeKTTYl)antémøCaelxeTalEdleKGackMNt;edayeRbI Laplaces’s
expansion. viFIenHeRbI determinant’s minor nig cofactor. FatunImYy² aij rbs;edETmINg;énlMdab;



n man minor M ij EdlCaedETmINg;lMdab; n − 1. RbsinebI minor RtUv)anKuNeday (− 1)i + j Edl
eKehAvafa cofactor rbs; aij enaH
Cij = (− 1)i + j M ij (A-10)

]TahrN_ eKmanedETmINg;lMdab;dI
a11 a12 a13
a21 a22 a23
a31 a32 a33

cofactor sRmab;FatuenAkñúgCYredkTImYyKW
a22 a23 a22 a23
C11 = (− 1)1+1 =
a32 a33 a32 a33
a21 a23 a a
C12 = (− 1)1+ 2 = − 21 23
a31 a33 a31 a33
a21 a22 a21 a22
C13 = (− 1)1+ 3 =
a31 a32 a31 a32

Laplace’s expansion sRmab;edETmINg;lMdab; n ¬smIkar A-9¦ erobrab;fatémøCaelxEdltMNag
eGayedETmINg;esμInwgplbUkénplKuNénFatuénCYredk b¤CYrQr nig cofactor EdlRtUvKñarbs;va
eBalKW
D = ai1Ci1 + ai 2Ci 2 + L + ainCin ( i = 1,2,K, n )
b¤ D = a1 jC1 j + a2 jC2 j + L + anjCnj ( j = 1,2, K , n ) (A-11)

sRmab;karGnuvtþ eyIgeXIjfaedaysar cofactor, edETmINg; D RtUv)ankMNt;eday n edETmINg;
lMdab; n − 1. eKGackMNt;edETmINg;bnþedayrUbmnþdUcKña b:uEnþvaRtUvkMNt; (n − 1) edETmINg;lMdab;
(n − 2) nigbnþbnÞab;. dMeNIrkarénkarKNnaenHbnþrhUtdl;edETmINg;EdlRtUvkarKNnaRtUv)ankat;
bnßyrhUtdl;lMdab;BIr cMENkÉ cofactor énFatuCaFatueTalrbs; D . ]TahrN_ eKmanedETmINg;
lMdab;TIBIrxageRkam
3 5
D=
−1 2
eyIgGacKNna D BIFatutamCYredkxagelIbMput EdleGay
D = 3(− 1)1+1 (2 ) + 5(− 1)1+ 2 (− 1) = 11
b¤ edayeRbIFatuénCYrQrTIBIr eyIg)an
D = 5(− 1)1+ 2 (− 1) + 2(− 1)2 + 2 (3) = 11



RbesIrCagkareRbIsmIkar A-11 eKGackMNt;edETmINg;lMdab;BIredayKuNFatutamGgát;RTUg
BIxagelIEpñkxageqVgeTAeRkamEpñkxagsþaM ehIydknwgplKuNénFatuGgát;RTUgBIxagelIEpñkxagsþaM
eTAxageRkamEpñkxageqVg GnuvtþtamsBaØaRBYj

BicarNaedETmINg;lMdab;bI
1 3 −1
D = 4 2 6
−1 0 2

edayeRbIsmIkar A-11 eyIgGacKNna D edayeRbIFatutambeNþayCYredkxagelI eyIg)an
2 6 4 6 4 2
D = (1)(− 1)1+1 + (3)(− 1)1+ 2 + (− 1)(− 1)1+3
0 2 −1 2 −1 0

= 1(4 − 0 ) − 3(8 + 6) − 1(0 + 2 ) = −40
eKk¾GacKNna D edayeRbIFatutambeNþayCYredkTIBIr.

A >$> cRmas;rbs;m:aRTIs (Inverse of a matrix)
BicarNasMNMuénsmIkarlIenEG‘rbIxageRkam
a11x1 + a12 x2 + a13 x3 = c1

a21x1 + a22 x2 + a23 x3 = c2

a31x1 + a32 x2 + a33 x3 = c3
EdlsresrkñúgTRmg;m:aRTIsdUcxageRkam
⎡ a11 a12 a13 ⎤ ⎡ x1 ⎤ ⎡ c1 ⎤
⎢a a ⎥⎢ ⎥ ⎢ ⎥
⎢ 21 22 a23 ⎥ ⎢ x2 ⎥ = ⎢c2 ⎥ (A-12)
⎣ a31 a32 a33 ⎥ ⎢ x3 ⎥ ⎢c3 ⎥
⎢ ⎦⎣ ⎦ ⎣ ⎦
b¤ Ax = C (A-13)

eKGacedaHRsayrk x edayEck C nwg A b:uEnþvaminmanRbmaNviFIEckenAkñúgm:aRTIseT. CMnYseday
RbmaNviFIEck eKeRbIm:aRTIsRcas. cRmas;rbs;m:aRTIs A Cam:aRTIsdéTeTotEdlmanlMdab;dUcKña
ehIyeKsresrvaCanimitþsBaØa A−1 . eKmanlkçN³dUcxageRkam
AA−1 = A−1 A = I
Edl I Cam:aRTIsÉktþa. KuNGgÁTaMgBIrénsmIkar A-13 eday A−1 eyIgTTYl)an


AA−1x = A−1C
edaysar A−1Ax = Ix = x eyIg)an
x = A−1C (A-14)
luHRtaEteKGacTTYl)an A eTIbeKGacedaHRsay x .
−1

sRmab;karKNnaedayéd viFIEdleRbI A−1 EdlRtUv)anbegáIteLIgedayeRbIc,ab; Cramer. enA
TIenHeyIgmin)anerobrab;BIkarbegáItvaeT eyIgnwgbgðajEtlT§plb:ueNÑaH. cMeBaHbBaðaejnH eKGac
sresrFatuenAkñúgm:aRTIsénsmIkar A-14 Ca
x = A−1C
⎡ x1 ⎤ ⎡C11 C21 C31 ⎤ ⎡ c1 ⎤
⎢x ⎥ = 1 ⎢C C ⎥⎢ ⎥ (A-15)
⎢ 2⎥ A ⎢ 12 22 C32 ⎥ ⎢c2 ⎥
⎢ x3 ⎥
⎣ ⎦ ⎢C13 C23 C33 ⎥ ⎢c3 ⎥
⎣ ⎦⎣ ⎦
enATIenH A CaedETmINg;rbs;m:aRTIs A EdlRtUv)ankMNt;edayeRbI Laplace expansion Edlerobrab;
enAkñúgkfaxNÐ A-3. m:aRTIskaerEdlman cofactor Cij RtUv)aneKeGayeQμaHfa adjoint matrix.
edaykareRbobeFob eyIgeXIjfaeKGacTTYlm:aRTIsRcas A−1 BIm:aRTIs A edaydMbUgeKRtUvCMnYsFatu
aij eday cofactor Cij bnÞab;mkeFVI transpose m:aRTIsEdlCalT§pl ¬EdleKeFVIeGay)an adjoint

matrix¦ ehIycugeRkayedayKuN adjoint matrix CamYynwg 1 / A .

edIm,IbgðajBIrebobedIm,ITTYl A−1 eyIgnwgBicarNadMeNaHRsayénRbB½n§smIkarlIenEG‘rxag
eRkam
x1 − x2 + x3 = −1
− x1 + x2 + x3 = −1 (A-16)
x1 + 2 x2 − 2 x3 = 5
⎡ 1 −1 1 ⎤
enATIenH A = ⎢− 1 1 1 ⎥
⎢ ⎥
⎢ 1 2 − 2⎥
⎣ ⎦
m:aRTIs cofactor sRmab; A KW
⎡ 1 1 −1 1 −1 1 ⎤
⎢ − ⎥
⎢ 2 −2 1 −2 1 2 ⎥
⎢ −1 1 1 1 1 −1 ⎥
C = ⎢− −
⎢ 2 −2 1 −2 1 2 ⎥⎥
⎢ −1 1 1 1 1 −1 ⎥
⎢ 1 1 −
⎣ −1 1 −1 1 ⎥ ⎦
KNnaedETmINg; adjoint matrix KW


⎡ − 4 0 − 2⎤
C = ⎢ − 1 − 3 − 2⎥
T
⎢ ⎥
⎢− 3 − 3 0 ⎥
⎣ ⎦
1 −1 1
edaysar A = − 1 1 1 = −6
1 2 −2

dUcenH cRmas;rbs; A KW
⎡ − 4 0 − 2⎤
= − ⎢ − 1 − 3 − 2⎥
−1 1⎢
A ⎥
6
⎢− 3 − 3 0 ⎥
⎣ ⎦
dMeNaHRsayénsmIkar A-16 eyIg)an
⎡ x1 ⎤ ⎡− 4 0 − 2⎤ ⎡− 1⎤
⎢ x ⎥ = − 1 ⎢ − 1 − 3 − 2⎥ ⎢− 1⎥
⎢ 2⎥ 6⎢ ⎥⎢ ⎥
⎢ x3 ⎥
⎣ ⎦ ⎢− 3 − 3 0 ⎥ ⎢ 5 ⎥
⎣ ⎦⎣ ⎦
x1 = − [(− 4)(− 1) + 0(− 1) + (− 2)(5)] = 1
1
6
x2 = − [(− 1)(− 1) + (− 3)(− 1) + (− 2)(5)] = 1
1
6
x3 = − [(− 3)(− 1) + (− 3)(− 1) + (0)(5)] = −1
1
6
eyIgeXIjy:agc,as;fa karKNnaCaelxTamTarkarBnøatsmIkary:agEvg. sRmab;mUlehtuenH eKeRbI
kMuBüÚT½rkñúgkarviPaKeRKagedIm,IedaHRsaycRmas;rbs;m:aRTIs.

A >%> viFI Gauss sRmab;KNnaRbB½n§smIkar
(The Gauss method for solving simultaneous equation)
enAeBleKRtUvkaredaHRsayRbB½n§smIkarlIenEG‘reRcIn eKGaceRbIviFIkat;bnßy Gause BIeRBaH
vamanRbsiT§PaBkñúgkaredaHRsay. karGnuvtþviFIenHTamTarkaredaHRsayrkGBaØatmYykñúgcMeNam n
smIkar eBalKW x1 edayeRbIGBaØatdéTeTot x2 , x3,..., xn . CMnYssmIkarEdleKehAfa pivotal
equation eTAkñúgsmIkarEdlenAsl;Edlman n − 1 smIkarCamYynwg n − 1 GBaØat. GnuvtþRbmaNviFI

enHbnþeTotedIm,IedaHRsayRbB½n§smIkarTaMgenHedIm,Irk x2 CaGnuKmn_eTAnwgGBaØatEdlenAsl; n − 2
GBaØat x3, x4 ,..., xn begáIt)an pivotal equation TIBIr. bnÞab;mkCMnYssmIkareTAkñúgsmIkardéTeTot
EdleFVIeGayenAsl; n − 3 smIkarCamYynwg n − 3 GBaØat. GnuvtþRbmaNviFIenHsareLIgvijrhUtTal;
EtenAsl; pivotal equation mYyEdlmanGBaØatmYy EdlbnÞab;mkeyIgnwgedaHRsayrkva. bnÞab;mk
eyIgGacedaHRsayrkGBaØatdéTeTotedayCMnYsvaeTAkñúg pivotal equation. edIm,IeFVIeGaydMeNaH


RsayenHmanlkçN³suRkit enAeBlbegáIt pivotal equation nImYy² eKRtUvEteRCIserIssmIkarEdl
manemKuNFMCageKedIm,Ikat;bnßyGBaØat. eyIgGacbgðajkaredaHRsayenHtamry³]TahrN_.
edaHRsayRbB½n§smIkarxageRkamedayeRbIviFI Gause:
− 2 x1 + 8 x2 + 2 x3 = 2 (A-17)
2 x1 − x2 + x3 = 2 (A-18)
4 x1 − 5 x2 + 3 x3 = 4 (A-19)
eyIgnwgcab;epþImedaykat;bnßy x1 . emKuNEdlFMbMputrbs; x1 KWenAkñúgsmIkar A-19 dUcenHeyIgnwg
eRbIvaCa pivotal equation. edaHRsayrk x1 eyIg)an
x1 = 1 + 1.25 x2 − 0.75 x3 (A-20)
edayCMnYseTAkñúgsmIkar A-17 nig A-18 nigedaysRmYl eyIg)an
2.75 x2 + 1.75 x3 = 2 (A-21)
1.5 x2 − 0.5 x3 = 0 (A-22)
bnÞab;mk eyIgkat;bnßy x2 . edayeRCIserIssmIkar A-21 sRmab; pivotal equation edaysarem-
KuN x2 mantémøFMCageKenATIenH eyIg)an
x2 = 0.727 − 0.636 x3 (A-23)
edayCMnYssmIkarenHeTAkñúgsmIkar A-22 nigedaysRmYlva eyIgTTYl)an pivotal equation cug
eRkay EdleyIgGacedaHRsayrk x3 . eyIgTTYl)an x3 = 0.75 . edayCMnYstémøenHeTAkñúg pivotal
equation A-23 eyIg)an x2 = 0.25 . cugeRkay BI pivotal equation A-20 eyIg)an x1 = 0.75 .



cMeNaT
2 − 3⎤ ⎡ − 1 3 − 2⎤
A-1 RbsinebI A = ⎡6
⎢
4
1 5⎥
ehIy B=⎢2 4 1 ⎥ . kMNt; AB .
⎣ ⎦ ⎢ ⎥
⎡ 4 − 1 0⎤ ⎢0 7 5 ⎥
B=⎢ ⎥ . kMNt; A + B nig A − 2B . ⎣ ⎦
⎣2 0 8⎦
A-11 RbsinebI A = ⎡1 5⎤ . kMNt; AAT .
2
⎢ 3⎥
A-2 RbsinebI A = [6 1 3] ehIy B = [1 6 3] ⎣ ⎦

bgðajfa ( A + B )T = AT + BT . A-12 bgðajfa A(B + C ) = AB + AC RbsinebI

⎡ 3 5⎤ ⎡3⎤ ⎡5⎤
A-3 RbsinebI A = ⎢ ⎥ kMNt; A + A .
T ⎡ 2 1 6⎤
A=⎢ /
B = ⎢ 1 ⎥ C = ⎢− 1⎥ / .
−2 7
⎣ ⎦ ⎣ 4 5 3⎥
⎦
⎢ ⎥ ⎢ ⎥
⎢ − 6⎥
⎣ ⎦ ⎢2⎥
⎣ ⎦
⎡1⎤
A-4 RbsinebI A = ⎢0 ⎥
⎢ ⎥ ehIy B = [2 − 1 3] A-13 bgðajfa A(BC ) = ( AB )C RbsinebI
⎢5⎥
⎣ ⎦ ⎡3⎤
⎡ 2 1 6⎤
kMNt; AB . A=⎢
4 5 3⎥
/
B = ⎢ 1 ⎥ C = [5
⎢ ⎥ / − 1 2] .
⎣ ⎦ ⎢ − 6⎥
⎣ ⎦
⎡ 6 2 2⎤ 1 3 5
A = ⎢− 5 1 1 ⎥
RbsinebI ehIy KNnaedETmINg; nig .
2 5
A-5 ⎢ ⎥ A-14 2 7 1
⎢ 0 3 1⎥ 7 1
⎣ ⎦ 3 8 6
⎡ − 1 3 1⎤
B = ⎢ 2 − 5 1⎥ AB kMNt; . A-15 RbsinebI A = ⎡5
⎢3
1⎤
− 2⎥
. kMNt; A−1 .
⎢ ⎥ ⎣ ⎦
⎢ 0 7 5⎥
⎣ ⎦ ⎡0 1 5 ⎤
A-6 kMNt; BA sRmab;m:aRTIséncMeNaT A-5. A-16 RbsinebI A = ⎢2 5 0⎥
⎢ ⎥ . kMNt; A−1 .
⎢1 − 1 2 ⎥
⎣ ⎦
⎡ 5 7⎤ ⎡6 ⎤
A-7 RbsinebI A = ⎢ ⎥ ehIy B = ⎢7⎥
⎣− 2 1 ⎦ ⎣ ⎦ A-17 edaHRsaysmIkar − x1 + 4 x2 + x3 = 1 /
kMNt; AB . 2 x1 − x2 + x3 = 2 ehIy 4 x1 − 5 x2 + 3x3 = 4
⎡3⎤
RbsinebI ⎡1 8 4⎤
ehIy edayeRbIsmIkarm:aRTIs X = A−1C .
A-8 A=⎢ ⎥ B=⎢ 2 ⎥
⎢ ⎥
⎣1 2 3⎦ ⎢ − 6⎥ A-18 edaHRsaysmIkarenAkñúgcMeNaT A-17
⎣ ⎦
kMNt; AB . edayeRbIviFI Gause.
⎡2 7 3⎤ A-19 edaHRsaysmIkar x1 − x2 + x3 = −1 /
A-9 RbsinebI A = ⎢ ehIy
−2⎣ 1 0⎥
⎦ − x1 + x2 + x3 = −1 ehIy x1 + 2 x2 − 2 x3 = 5
⎡6⎤
B=⎢9⎥
⎢ ⎥ kMNt; AB . edayeRbIsmIkarm:aRTIs X = A−1B .
⎢− 1⎥
⎣ ⎦ A-20 edaHRsaysmIkarenAkñúgcMeNaT A-19
⎡6 4 2⎤
A-10 RbsinebI A = ⎢2 1 1 ⎥ ehIy edayeRbIviFI Gause.
⎢ ⎥
⎢0 − 3 1 ⎥
⎣ ⎦
Problems T.Chhay -549

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