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# A.matrix algebra for structural analysisdoc

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### A.matrix algebra for structural analysisdoc

1. 1. 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
2. 2. 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
3. 3. Department of Civil Engineering NPIC 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 m:aRTIssRmab;karviPaKeRKOgbgÁúM T.Chhay -540
4. 4. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 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 Matrix algebra for structural analysis T.Chhay -541
5. 5. Department of Civil Engineering NPIC ⎡ 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 m:aRTIssRmab;karviPaKeRKOgbgÁúM T.Chhay -542
6. 6. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 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; Matrix algebra for structural analysis T.Chhay -543
7. 7. Department of Civil Engineering NPIC 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 m:aRTIssRmab;karviPaKeRKOgbgÁúM T.Chhay -544
8. 8. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 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 Matrix algebra for structural analysis T.Chhay -545
9. 9. Department of Civil Engineering NPIC 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 m:aRTIssRmab;karviPaKeRKOgbgÁúM T.Chhay -546
10. 10. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ⎡ − 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 Matrix algebra for structural analysis T.Chhay -547
11. 11. Department of Civil Engineering NPIC 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 . m:aRTIssRmab;karviPaKeRKOgbgÁúM T.Chhay -548
12. 12. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 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