{"text_format":"3","workbook":"xlsx","n_sheets":4,"sheets":["Beispiel_1","Beispiel_2","Beispiel_3","Beispiel_fuer_Regel_3"],"not_rendered":{"images":1},"read":"whole","date_system":"1900"}

{"sheet":"Beispiel_1","index":0,"n_rows":29,"n_cols":12,"rows_kept":29}
["Gaussches Eliminationsverfahren mit Hilfe der Multiplikation von Elementarmatrizen"]
[]
[]
["Regeln: ","1) Die Multiplikation von links her mit einer Diagonalmatrix mit Einsen ausser in der Zelle ii, wo b steht, bewirkt eine Muliplikation der i-ten Zeile mit b."]
[null,"2) Eine Multiplikation mit einer Matrix (von links her) mit Einsen auf der Diagonalen und b in der Zelle ij bewirkt die "]
[null,"     Multiplikation der Zeile j mit b und deren Addition zur Zeile i. "]
[null,"3) Eine Multiplikation mit einer Matrix (von links her), die Einsen auf der Diagonalmatrix enthält ausser "]
[null,"     an den Stellen ii und jj, wo 0 steht, und die in den Zellen ij und ji 1 enthält, bewirkt ein Vertauschen der Zeilen i und j."]
[null,"(Die Gültigkeit der Regeln macht man sich am besten mit konkreten Beispielen klar, indem man die Multiplikation mit"]
[null,"den entsprechenden Matrizen durchrechnet. )"]
[]
["Die Matrizen, mit denen man von links her multipliziert, werden Elementarmatrizen genannt. "]
[]
["Beispiel:"]
["Gleichungssystem:",null,"2x1+  3x2=  5"]
[null,null,"3x1+15x2= 7"]
[]
[null,"Elementarmatrix",null,"Zu verwandelnde Matrizen",{"$nulls":3},"Jeweiliges Resultat",null,"(durch Multiplikation der jeweiligen Elementarmatrix mit der zu verwandelnden Matrix)"]
[null,{"$f":"1/D19","v":0.5},0,2,3,5,null,{"$f":"MMULT(B19:C20,D19:F20)","v":1},1.5,2.5,null,"(Es wird 0.5 =1/2 gewählt, da 0.5*2=1 gibt)"]
[null,0,1,3,15,7,null,3,15,7]
[]
[null,1,0,{"$f":"H19","v":1},{"$f":"I19","v":1.5},2.5,null,{"$f":"MMULT(B22:C23,D22:F23)","v":1},1.5,2.5]
[null,{"$f":"-H20","v":-3},1,{"$f":"H20","v":3},{"$f":"I20","v":15},{"$f":"J20","v":7},null,0,10.5,-0.5]
[]
[null,1,0,{"$f":"H22","v":1},{"$f":"I22","v":1.5},2.5,null,{"$f":"MMULT(B25:C26,D25:F26)","v":1},1.5,2.5]
[null,0,{"$f":"1/I23","v":0.09523809523809523},{"$f":"H23","v":0},{"$f":"I23","v":10.5},{"$f":"J23","v":-0.5},null,0,1,-0.047619047619047616]
[{"$nulls":8},"Lösung:"]
[null,1,{"$f":"-I25","v":-1.5},{"$f":"H25","v":1},{"$f":"I25","v":1.5},2.5,null,{"$f":"MMULT(B28:C29,D28:F29)","v":1},0,2.5714285714285716,"=x1"]
[null,0,1,{"$f":"H26","v":0},{"$f":"I26","v":1},{"$f":"J26","v":-0.047619047619047616},null,0,1,-0.047619047619047616,"=x2"]

{"sheet":"Beispiel_2","index":1,"n_rows":30,"n_cols":13,"rows_kept":30}
["Beispiel 2",null,"15x1+2x2+35x3=  5"]
[null,null,"28x1+11x2+12x3=12"]
[null,null,"1.5x1+5x2+0x3=14"]
[]
["(Die Regel 2 kann auf verschiedene Zeilen auf einmal angewendet werden, was das Verfahren beschleunigt)."]
[]
[null,"Elementarmatrixzen",null,null,"Zu verwandelnde Matrixzen",{"$nulls":3},"Ergebnisse"]
[null,{"$f":"1/E8","v":0.06666666666666667},0,0,15,2,35,5,{"$f":"MMULT(B8:D10,E8:H10)","v":1},0.13333333333333333,2.3333333333333335,0.3333333333333333]
[null,0,1,0,28,11,12,12,28,11,12,12]
[null,0,0,1,1.5,5,0,14,1.5,5,0,14]
[]
[null,1,0,0,{"$f":"I8","v":1},{"$f":"J8","v":0.13333333333333333},2.3333333333333335,0.3333333333333333,{"$f":"MMULT(B12:D14,E12:H14)","v":1},0.13333333333333333,2.3333333333333335,0.3333333333333333]
[null,{"$f":"-I9","v":-28},1,0,{"$f":"I9","v":28},11,12,12,0,7.266666666666667,-53.33333333333334,2.666666666666668]
[null,{"$f":"-I10","v":-1.5},0,1,1.5,5,0,14,0,4.8,-3.5,13.5]
[]
[null,1,0,0,{"$f":"I12","v":1},{"$f":"J12","v":0.13333333333333333},2.3333333333333335,0.3333333333333333,{"$f":"MMULT(B16:D18,E16:H18)","v":1},0.13333333333333333,2.3333333333333335,0.3333333333333333]
[null,0,{"$f":"1/J13","v":0.13761467889908258},0,{"$f":"I13","v":0},7.266666666666667,-53.33333333333334,2.666666666666668,0,1,-7.339449541284405,0.36697247706422037]
[null,0,0,1,0,4.8,-3.5,13.5,0,4.8,-3.5,13.5]
[]
[null,1,{"$f":"-J16","v":-0.13333333333333333},0,{"$f":"I16","v":1},{"$f":"J16","v":0.13333333333333333},2.3333333333333335,0.3333333333333333,{"$f":"MMULT(B20:D22,E20:H22)","v":1},0,3.3119266055045875,0.2844036697247706]
[null,0,1,0,{"$f":"I17","v":0},1,-7.339449541284405,0.36697247706422037,0,1,-7.339449541284405,0.36697247706422037]
[null,0,{"$f":"-J18","v":-4.8},1,0,4.8,-3.5,13.5,0,0,31.72935779816514,11.738532110091743]
[]
[null,1,0,0,{"$f":"I20","v":1},{"$f":"J20","v":0},3.3119266055045875,0.2844036697247706,{"$f":"MMULT(B24:D26,E24:H26)","v":1},0,3.3119266055045875,0.2844036697247706]
[null,0,1,0,{"$f":"I21","v":0},1,-7.339449541284405,0.36697247706422037,0,1,-7.339449541284405,0.36697247706422037]
[null,0,0,{"$f":"1/K22","v":0.03151655341911233},0,0,31.72935779816514,11.738532110091743,0,0,1,0.3699580743096718]
[{"$nulls":9},"Lösung:"]
[null,1,0,{"$f":"-K24","v":-3.3119266055045875},{"$f":"I24","v":1},{"$f":"J24","v":0},3.3119266055045875,0.2844036697247706,{"$f":"MMULT(B28:D30,E28:H30)","v":1},0,0,-0.9408703195026747,"=x1"]
[null,0,1,{"$f":"-K25","v":7.339449541284405},{"$f":"I25","v":0},1,-7.339449541284405,0.36697247706422037,0,1,0,3.0822610958508028,"=x2"]
[null,0,0,1,0,0,1,0.3699580743096718,0,0,1,0.3699580743096718,"=x3"]

{"sheet":"Beispiel_3","index":2,"n_rows":52,"n_cols":16,"rows_kept":52}
["Beispiel 3"]
{"$rows_empty":11}
[null,"Elementarmatrizen",{"$nulls":3},"Zu verändernde Matrizen",{"$nulls":4},"Ergebnisse"]
[null,{"$f":"1/F14","v":0.2},0,0,0,5,6,7,1,18,{"$f":"MMULT(B14:E17,F14:J17)","v":1},1.2000000000000002,1.4000000000000001,0.2,3.6]
[null,0,1,0,0,3,16,4,3,19,3,16,4,3,19]
[null,0,0,1,0,1,8,2,9,8,1,8,2,9,8]
[null,0,0,0,1,1.2,14,2,20,17,1.2,14,2,20,17]
[]
[null,1,0,0,0,{"$f":"K14","v":1},{"$f":"L14","v":1.2000000000000002},1.4000000000000001,0.2,3.6,{"$f":"MMULT(B19:E22,F19:J22)","v":1},1.2000000000000002,1.4000000000000001,0.2,3.6]
[null,{"$f":"-K15","v":-3},1,0,0,{"$f":"K15","v":3},{"$f":"L15","v":16},{"$f":"M15","v":4},{"$f":"N15","v":3},{"$f":"O15","v":19},0,12.399999999999999,-0.20000000000000018,2.4,8.2]
[null,{"$f":"-K16","v":-1},0,1,0,1,8,2,9,8,0,6.8,0.5999999999999999,8.8,4.4]
[null,-1.2,0,0,1,1.2,14,2,20,17,0,12.56,0.31999999999999984,19.76,12.68]
[]
[null,1,0,0,0,{"$f":"K19","v":1},{"$f":"L19","v":1.2000000000000002},1.4000000000000001,0.2,3.6,{"$f":"MMULT(B24:E27,F24:J27)","v":1},1.2000000000000002,1.4000000000000001,0.2,3.6]
[null,0,{"$f":"1/L20","v":0.08064516129032259},0,0,{"$f":"K20","v":0},{"$f":"L20","v":12.399999999999999},{"$f":"M20","v":-0.20000000000000018},{"$f":"N20","v":2.4},{"$f":"O20","v":8.2},0,1,-0.016129032258064533,0.19354838709677422,0.6612903225806452]
[null,0,0,1,0,0,6.8,0.5999999999999999,8.8,4.4,0,6.8,0.5999999999999999,8.8,4.4]
[null,0,0,0,1,0,12.56,0.31999999999999984,19.76,12.68,0,12.56,0.31999999999999984,19.76,12.68]
[]
[null,1,{"$f":"-L24","v":-1.2000000000000002},0,0,{"$f":"K24","v":1},{"$f":"L24","v":1.2000000000000002},1.4000000000000001,0.2,3.6,{"$f":"MMULT(B29:E32,F29:J32)","v":1},0,1.4193548387096775,-0.03225806451612909,2.8064516129032255]
[null,0,1,0,0,{"$f":"K25","v":0},{"$f":"L25","v":1},{"$f":"M25","v":-0.016129032258064533},{"$f":"N25","v":0.19354838709677422},{"$f":"O25","v":0.6612903225806452},0,1,-0.016129032258064533,0.19354838709677422,0.6612903225806452]
[null,0,{"$f":"-L26","v":-6.8},1,0,0,6.8,0.5999999999999999,8.8,4.4,0,0,0.7096774193548387,7.483870967741936,-0.09677419354838701]
[null,0,{"$f":"-L27","v":-12.56},0,1,0,12.56,0.31999999999999984,19.76,12.68,0,0,0.5225806451612904,17.32903225806452,4.374193548387096]
[]
[null,1,0,0,0,{"$f":"K29","v":1},{"$f":"L29","v":0},1.4193548387096775,-0.03225806451612909,2.8064516129032255,{"$f":"MMULT(B34:E37,F34:J37)","v":1},0,1.4193548387096775,-0.03225806451612909,2.8064516129032255]
[null,0,1,0,0,{"$f":"K30","v":0},{"$f":"L30","v":1},{"$f":"M30","v":-0.016129032258064533},{"$f":"N30","v":0.19354838709677422},{"$f":"O30","v":0.6612903225806452},0,1,-0.016129032258064533,0.19354838709677422,0.6612903225806452]
[null,0,0,{"$f":"1/M31","v":1.409090909090909},0,0,0,0.7096774193548387,7.483870967741936,-0.09677419354838701,0,0,1,10.545454545454545,-0.13636363636363624]
[null,0,0,0,1,0,0,0.5225806451612904,17.32903225806452,4.374193548387096,0,0,0.5225806451612904,17.32903225806452,4.374193548387096]
[]
[null,1,0,{"$f":"-M34","v":-1.4193548387096775},0,{"$f":"K34","v":1},{"$f":"L34","v":0},1.4193548387096775,-0.03225806451612909,2.8064516129032255,{"$f":"MMULT(B39:E42,F39:J42)","v":1},0,0,-15.000000000000002,2.9999999999999996]
[null,0,1,{"$f":"-M35","v":0.016129032258064533},0,{"$f":"K35","v":0},{"$f":"L35","v":1},{"$f":"M35","v":-0.016129032258064533},{"$f":"N35","v":0.19354838709677422},{"$f":"O35","v":0.6612903225806452},0,1,0,0.36363636363636387,0.6590909090909092]
[null,0,0,1,0,0,0,1,10.545454545454545,-0.13636363636363624,0,0,1,10.545454545454545,-0.13636363636363624]
[null,0,0,{"$f":"-M37","v":-0.5225806451612904},1,0,0,0.5225806451612904,17.32903225806452,4.374193548387096,0,0,0,11.81818181818182,4.445454545454544]
[]
[null,1,0,0,0,{"$f":"K39","v":1},{"$f":"L39","v":0},0,-15.000000000000002,2.9999999999999996,{"$f":"MMULT(B44:E47,F44:J47)","v":1},0,0,-15.000000000000002,2.9999999999999996]
[null,0,1,0,0,{"$f":"K40","v":0},{"$f":"L40","v":1},{"$f":"M40","v":0},{"$f":"N40","v":0.36363636363636387},{"$f":"O40","v":0.6590909090909092},0,1,0,0.36363636363636387,0.6590909090909092]
[null,0,0,1,0,0,0,1,10.545454545454545,-0.13636363636363624,0,0,1,10.545454545454545,-0.13636363636363624]
[null,0,0,0,{"$f":"1/N42","v":0.0846153846153846},0,0,0,11.81818181818182,4.445454545454544,0,0,0,1,0.376153846153846]
[{"$nulls":11},"Lösung"]
[null,1,0,0,{"$f":"-N44","v":15.000000000000002},{"$f":"K44","v":1},{"$f":"L44","v":0},0,-15.000000000000002,2.9999999999999996,{"$f":"MMULT(B49:E52,F49:J52)","v":1},0,0,0,8.642307692307691,"=x1"]
[null,0,1,0,{"$f":"-N45","v":-0.36363636363636387},{"$f":"K45","v":0},{"$f":"L45","v":1},{"$f":"M45","v":0},{"$f":"N45","v":0.36363636363636387},{"$f":"O45","v":0.6590909090909092},0,1,0,0,0.5223076923076924,"=x2"]
[null,0,0,1,-10.545454545454545,0,0,1,10.545454545454545,-0.13636363636363624,0,0,1,0,-4.103076923076921,"=x3"]
[null,0,0,0,1,0,0,0,1,0.376153846153846,0,0,0,1,0.376153846153846,"=x4"]

{"sheet":"Beispiel_fuer_Regel_3","index":3,"n_rows":35,"n_cols":14,"rows_kept":35}
["Regel 3: Vertrauschen von Zeilen (bei den ersten drei Beispielen nie gebraucht; Achtung: in jeder Zeile "]
["und in jeder Spalte der Elementarmatrizen hat es je eine Eins, die Matrizen sind symmetrisch)"]
[]
["Vertauschen von Zeile 1 mit Zeile 3:"]
["Elementarmatrix",{"$nulls":3},"Zu transformierende Matrix",{"$nulls":4},"Ergebnis"]
[0,0,1,0,5,6,7,1,18,{"$f":"MMULT(A6:D9,E6:I9)","v":1},8,2,9,8]
[0,1,0,0,3,16,4,3,19,3,16,4,3,19]
[1,0,0,0,1,8,2,9,8,5,6,7,1,18]
[0,0,0,1,1.2,14,2,20,17,1.2,14,2,20,17]
[]
[]
["Vertrauschen von Zeile 2 mit 3:"]
[1,0,0,0,5,6,7,1,18,{"$f":"MMULT(A13:D16,E13:I16)","v":5},6,7,1,18]
[0,0,1,0,3,16,4,3,19,1,8,2,9,8]
[0,1,0,0,1,8,2,9,8,3,16,4,3,19]
[0,0,0,1,1.2,14,2,20,17,1.2,14,2,20,17]
[]
[]
["Vertauschen von Zeile 1 mit 4"]
[0,0,0,1,5,6,7,1,18,{"$f":"MMULT(A20:D23,E20:I23)","v":1.2},14,2,20,17]
[0,1,0,0,3,16,4,3,19,3,16,4,3,19]
[0,0,1,0,1,8,2,9,8,1,8,2,9,8]
[1,0,0,0,1.2,14,2,20,17,5,6,7,1,18]
[]
["Vertauschen der Zeile 3 mit 4"]
[1,0,0,0,5,6,7,1,18,{"$f":"MMULT(A26:D29,E26:I29)","v":5},6,7,1,18]
[0,1,0,0,3,16,4,3,19,3,16,4,3,19]
[0,0,0,1,1,8,2,9,8,1.2,14,2,20,17]
[0,0,1,0,1.2,14,2,20,17,1,8,2,9,8]
[]
["Vertauschen der Zeile 2 mit 4"]
[1,0,0,0,5,6,7,1,18,{"$f":"MMULT(A32:D35,E32:I35)","v":5},6,7,1,18]
[0,0,0,1,3,16,4,3,19,1.2,14,2,20,17]
[0,0,1,0,1,8,2,9,8,1,8,2,9,8]
[0,1,0,0,1.2,14,2,20,17,3,16,4,3,19]