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[{"$nulls":3},"Eigenvalues of the Powers"]
[{"$nulls":3},"            of a Matrix"]
[]
[{"$nulls":3},"      This DIGMath module explores"]
[{"$nulls":3},"     the eigenvalues for the powers"]
[{"$nulls":3},"                 of a 2-by-2 matrix"]
[]
[]
[]
["   Define a non-singular matrix A =   ",{"$nulls":3},"using the sliders below:"]
[]
[{"$nulls":4},{"$f":"D16","v":2.7},{"$f":"D19","v":1}]
[{"$nulls":4},{"$f":"D22","v":4},{"$f":"D25","v":2}]
[{"$nulls":3},{"$f":"IF(a*d-b*cc=0,\"         This matrix is singular !\",\"\")","v":""}]
["Enter the value for a (-5 to 5):  ",{"$nulls":5}," "]
[null,null,"a =",{"$f":"(F16-50)/10","v":2.7},62,77]
[{"$nulls":9},{"$f":"\"The matrix A has \"&IF(Disc>0,\"two distinct eigenvalues\",IF(Disc=0,\"a double real eigenvalue\",\"a pair of complex eigenvalues\"))","v":"The matrix A has two distinct eigenvalues"},{"$nulls":7},{"$f":"\"The matrix A^2 has \"&IF(Disc00>0,\"two distinct eigenvalues\",IF(Disc00=0,\"a double real eigenvalue\",\"a pair of complex eigenvalues\"))","v":"The matrix A^2 has two distinct eigenvalues"},{"$nulls":6},{"$f":"\"The matrix A^3 has \"&IF(Disc000>0,\"two distinct eigenvalues\",IF(Disc000=0,\"a double real eigenvalue\",\"a pair of complex eigenvalues\"))","v":"The matrix A^3 has two distinct eigenvalues"}]
["Enter the value for b (-5 to 5):  ",{"$nulls":9},"l1 =",{"$f":"IF(Disc01>0,lambda01,IF(Disc01=0,lambda03,alpha05))","v":{"v":4.380394050424695,"d":"4.38"}},{"$f":"IF(AND(Disc<0,beta5>0),\" + \",IF(AND(Disc<0,beta5<0),\" - \",\"\"))","v":""},{"$f":"IF(Disc>=0,\"\",ABS(beta5))","v":""},{"$f":"IF(Disc<0,\"i\",\"\")","v":""},{"$nulls":3},"l1 =",{"$f":"IF(Disc00>0,lambda001,IF(Disc00=0,lambda003,alpha005))","v":{"v":19.18785203699607,"d":"19.19"}},{"$f":"IF(AND(Disc<0,beta5>0),\" + \",IF(AND(Disc<0,beta5<0),\" - \",\"\"))","v":""},{"$f":"IF(Disc>=0,\"\",ABS(beta5))","v":""},{"$nulls":3},"l1 =",{"$f":"IF(Disc000>0,lambda0001,IF(Disc000=0,lambda0003,alpha0005))","v":{"v":84.05035290328695,"d":"84.05"}}]
[null,null,"b =",{"$f":"(F19-50)/10","v":1},62,60,{"$nulls":4},"l2 =",{"$f":"IF(Disc01>0,lambda02,IF(Disc01=0,lambda04,alpha06))","v":{"v":0.31960594957530475,"d":"0.32"}},{"$f":"IF(AND(Disc<0,beta6>0),\" + \",IF(AND(Disc<0,beta6<0),\" - \",\"\"))","v":""},{"$f":"IF(Disc>=0,\"\",ABS(beta6))","v":""},{"$f":"IF(Disc<0,\"i\",\"\")","v":""},{"$nulls":3},"l2 =",{"$f":"IF(Disc00>0,lambda002,IF(Disc00=0,lambda004,alpha006))","v":{"v":0.10214796300393125,"d":"0.10"}},{"$f":"IF(AND(Disc<0,beta6>0),\" + \",IF(AND(Disc<0,beta6<0),\" - \",\"\"))","v":""},{"$f":"IF(Disc>=0,\"\",ABS(beta6))","v":""},{"$nulls":3},"l2 =",{"$f":"IF(Disc000>0,lambda0002,IF(Disx000=0,lambda0004,alpha0006))","v":{"v":0.032647096713048995,"d":"0.03"}}]
[]
["Enter the value for c (-5 to 5):  ",{"$nulls":8},"The characteristic equation for A  is",{"$nulls":7},"The characteristic equation for A^2 is",{"$nulls":6},"The characteristic equation for A^3 is"]
[null,null,"c =",{"$f":"(F22-50)/10","v":4},62,90,{"$nulls":4},"l2",{"$f":"IF(a0+d0 <=0,\" + \",\" - \")","v":" - "},{"$f":"ABS(a0+d0)&\"l\"","v":"4.7l"},{"$f":"IF(a0*d0 -b0*c0 >=0,\" +  \",\" - \")  & TEXT(ABS(a0*d0-b0*c0),\"0.0\")&\"  =  0\"","v":" +  1.4  =  0"},{"$nulls":4},"l2",{"$f":"IF(a00+d00 <=0,\" + \",\" - \")","v":" - "},{"$f":"ABS(a00+d00)&\"l\"","v":"19.29l"},{"$f":"IF(a00*d00 -b00*c00 >=0,\" +  \",\" - \")  & TEXT(ABS(a00*d00-b00*c00),\"0.00\")&\"  =  0\"","v":" +  1.96  =  0"},null,null,"l2",{"$f":"IF(a000+d000 <=0,\" + \",\" - \")","v":" - "},{"$f":"ABS(a000+d000)&\"l\"","v":"84.083l"},{"$f":"IF(a000*d000 -b000*c000 >=0,\" +  \",\" - \")  & TEXT(ABS(a000*d000-b000*c000),\"0.00\")&\"  =  0\"","v":" +  2.74  =  0"}]
[{"$nulls":9},"The discriminant = ",null,null,{"$f":"Disc01","v":{"v":16.490000000000002,"d":"16.49"}},{"$nulls":4},"   The discriminant = ",{"$nulls":3},{"$f":"Disc00","v":{"v":364.2641,"d":"364.26"}},null,null,"   The discriminant = ",null,null,{"$f":"Disc000","v":{"v":7058.974889000001,"d":"7058.97"}}]
["Enter the value for d (-5 to 5):  "]
[null,null,"d =",{"$f":"(F25-50)/10","v":2},62,70,{"$nulls":3},"The associated eigenvectors are",{"$nulls":7},"The associated eigenvectors are",{"$nulls":6},"The associated eigenvectors are"]
[]
[{"$nulls":10},"E1 =",{"$f":"IF(Disc<0,(TEXT(Sheet2!B31,\"0.00\")&IF(_EEE1>0,\" + \",\" - \")&TEXT(ABS(_EEE1),\"0.00\"))&\"i\",x0)","v":1},null,null,"E2 = ",{"$f":"IF(Disc<0,(TEXT(Sheet2!B34,\"0.00\")&IF(_EEE2>0,\" + \",\" - \")&TEXT(ABS(_EEE2),\"0.00\")&\"i\"),x00)","v":1}]
["Next, select a small variation x for c and d ",{"$nulls":10},{"$f":"IF(Disc<0,(TEXT(Sheet2!B32,\"0.00\")&IF(_FFF1>0,\" + \",\" - \")&TEXT(ABS(_FFF1),\"0.00\")&\"i\"),TEXT(y0,\"0.00\"))","v":"1.68"},{"$nulls":3},{"$f":"IF(Disc<0,(TEXT(Sheet2!H32,\"0.00\")&IF(_FFF2>0,\" + \",\" - \")&TEXT(ABS(_FFF2),\"0.00\")&\"i\"),TEXT(y00,\"0.00\"))","v":"-2.38"}]
[null,null,"x =",{"$f":"(F29-100)/100","v":0.26},62,126]
[]
["The nearly singular matrix is:"]
[{"$nulls":4},{"$f":"a","v":2.7},{"$f":"b","v":1},{"$nulls":3},{"$f":"MINVERSE(E32:F33)","v":{"v":1.226927252985885,"d":"1.23"}},-0.5428881650380024]
[{"$nulls":4},{"$f":"cc+xx0","v":4.26},{"$f":"d+xx0","v":2.26},{"$nulls":3},{"v":-2.31270358306189,"d":"-2.31"},1.4657980456026065]
[]
[]
["Click each item below for suggestions and investigations"]
[{"$nulls":3},"   Item 1"]
[{"$nulls":3},"   Item 2",{"$nulls":4}," "]
[{"$nulls":3},"   Item 3"]
[{"$nulls":3},"   Item 4"]
[]
[]
[]
[]
[]
["To access/download all the DIGMath Apps for Differential Equations and Matrix Algebra"]
["    Click this link:",null,null,"https://we-dig-math.org/dynamic-differential-equations.html"]
[]
["Created by:  Sheldon P. Gordon   "]
["                  Farmingdale State College "]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Sheet2","index":1,"n_rows":54,"n_cols":32,"rows_kept":54}
[{"$f":"Sheet1!E12","v":2.7},2,null,"plotting points:",null,0,0,null,null,0,0,null,{"$f":"ax","v":2.7},{"$f":"bx","v":1},null,"plotting points:",null,0,0,null,0,0,null,"A =",{"$f":"Sheet1!E12:F13","v":2.7},1,{"$nulls":4},"graphing points"]
[{"$f":"Sheet1!E13","v":4},{"$f":"Sheet1!F13","v":2},{"$nulls":3},{"$f":"IF(Disc>=0,x0,0)","v":1},{"$f":"IF(Disc>=0,y0,0)","v":1.6803940504246953},null,null,{"$f":"IF(Disc>=0,x00,0)","v":1},{"$f":"IF(Disc>=0,y00,0)","v":-2.3803940504246954},null,{"$f":"cx","v":4.26},{"$f":"dx","v":2.26},{"$nulls":3},{"$f":"IF(Discx>=0,xx0,0)","v":0.26},{"$f":"IF(Discx>=0,y0,0)","v":1.6803940504246953},null,{"$f":"IF(Disc>=0,x00,0)","v":1},{"$f":"IF(Disc>=0,y00,0)","v":-2.3803940504246954},null,null,4,2,{"$nulls":4},"x","y"]
[{"$nulls":23},"Disc =",{"$f":"(a0+d0)^2-4*(a0*d0-b0*c0)","v":16.490000000000002},null,"l =",{"$f":"eigVal(_xlfn.ANCHORARRAY(Y1))","v":{"$error":"#NAME?"}},null,null,{"$f":"IF(Disc01>0,(lambda01+lambda02)/2-3,IF(Disc01=0,lambda03-2,(a0+b0)/2-2))","v":-0.6499999999999999},{"$f":"AE3^2-(a0+d0)*AE3+(a0*d0-b0*c0)","v":4.8774999999999995}]
["Disc =",{"$f":"(a+d)^2-4*(a*d-b*cc)","v":16.490000000000002},null,"vector points",null,0,{"$f":"G1","v":0},{"$nulls":5},"Disc =",{"$f":"(ax+dx)^2-4*(ax*dx-bx*cx)","v":17.233600000000003},null,"vector points",null,0,{"$f":"S1","v":0},{"$nulls":11},{"$f":"IF(Disc01>0,(lambda02+lambda01)/2+3,IF(Disc01=0,lambda03+2,(a0+b0)/2+2))","v":5.35},4.877499999999996]
[{"$nulls":5},0,{"$f":"G2","v":1.6803940504246953},{"$nulls":10},0,{"$f":"S2","v":1.6803940504246953},{"$nulls":3},"D > 0","lambda1 =",{"$f":"IF(Disc01>0,((a0+d0+SQRT(Disc))/2),\"\")","v":{"v":4.380394050424695,"d":"4.380"}},{"$nulls":5},{"$f":"(AE3+AE4)/2","v":2.3499999999999996},-4.1225]
["D > 0",{"$nulls":11},"D > 0",{"$nulls":10},"lambda2 =",{"$f":"IF(Disc01>0,((a0+d0-SQRT(Disc01))/2),\"\")","v":{"v":0.31960594957530475,"d":"0.320"}},{"$nulls":5},{"$f":"IF(Disc01>0,lambda01,IF(Disc01=0,lambda03,\"\"))","v":4.380394050424695},1.7763568394002505E-15]
["lambda1 =",{"$f":"IF(Disc>0,((a+d+SQRT(Disc))/2),\"\")","v":{"v":4.380394050424695,"d":"4.380"}},{"$nulls":3},"theta0 =",{"$f":"ATAN((G2-G1)/(F2-F1))","v":1.0339886559963507},null,{"$f":"F2-F1","v":1},{"$nulls":3},"lambdax1 =",{"$f":"IF(Discx>0,((ax+dx+SQRT(Discx))/2),\"\")","v":{"v":4.555668566992332,"d":"4.556"}},{"$nulls":3},"theta0 =",{"$f":"ATAN((S2-S1)/(R2-R1))","v":1.4172879870397999},null,{"$f":"R2-R1","v":0.26},null,"eigenvects:",{"$f":"IF(b0<>0,1,IF(AND(b0=0,c0<>0),(lambda01-d0)/c0,IF(AND(b0=0,c0=0),0,\"\")))","v":1},{"$f":"IF(b0<>0,1,IF(AND(b0=0,c0<>0),(lambda02-d0)/c0,IF(AND(b0=0,c0=0),0,\"\")))","v":1},{"$nulls":5},{"$f":"IF(Disc01>0,lambda01,IF(Disc01=0,lambda04,\"\"))","v":4.380394050424695},1.7763568394002505E-15]
["lambda2 =",{"$f":"IF(Disc>0,((a+d-SQRT(Disc))/2),\"\")","v":{"v":0.31960594957530475,"d":"0.320"}},{"$nulls":3},"theta1 =",{"$f":"ATAN((K2-K1)/(J2-J1))","v":-1.173084597478422},null,{"$f":"J2-J1","v":1},{"$nulls":3},"lambdax2 =",{"$f":"IF(Discx>0,((ax+dx-SQRT(Discx))/2),\"\")","v":{"v":0.40433143300766794,"d":"0.404"}},{"$nulls":3},"theta1 =",{"$f":"ATAN((V2-V1)/(U2-U1))","v":-1.173084597478422},null,{"$f":"U2-U1","v":1},null,null,{"$f":"IF(b0<>0,(lambda01-a0)/b0,IF(AND(b0=0,c0<>0),1,IF(AND(b0=0,c0=0),0,\"\")))","v":1.6803940504246953},{"$f":"IF(b0<>0,(lambda02-a0)/b0,IF(AND(b0=0,c0<>0),1,IF(AND(b0=0,c0=0),0,\"\")))","v":{"v":-2.3803940504246954,"d":"-2.380"}},{"$nulls":5},{"$f":"IF(Disc01>0,lambda02,IF(Disc01=0,lambda04,\"\"))","v":0.31960594957530475},0]
[{"$nulls":22},"D = 0","lambda3 =",{"$f":"IF(Disc01=0,((a0+d0)/2),\"\")","v":""}]
["eigenvectors:",{"$nulls":11},"eigenvectors:",{"$nulls":10},"lambda4 =",{"$f":"IF(Disc01=0,((a0+d0)/2),\"\")","v":""}]
[null,{"$f":"IF(b<>0,1,IF(AND(b=0,cc<>0),(lambda1-d)/cc,IF(AND(b=0,cc=0),0,\"\")))","v":1},{"$f":"IF(b<>0,1,IF(AND(b=0,cc<>0),(lambda2-d)/cc,IF(AND(b=0,cc=0),0,\"\")))","v":1},{"$nulls":10},{"$f":"IF(bx<>0,1,IF(AND(bx=0,cx<>0),(lambdax1-dx)/cx,IF(AND(bx=0,cx=0),0,\"\")))","v":1},{"$f":"IF(bx<>0,1,IF(AND(bx=0,cx<>0),(lambdax2-d)/cx,IF(AND(bx=0,cx=0),0,\"\")))","v":1},{"$nulls":7},"eigenvects:",{"$f":"IF(b0<>0,1,IF(AND(b0=0,c0<>0),(lambda03-d0)/c0,IF(AND(b0=0,c0=0),0)))","v":1},{"$f":"IF(b0<>0,1,IF(AND(b0=0,c0<>0),(lambda04-d0)/c0,IF(AND(b0=0,c0=0),0)))","v":1}]
[null,{"$f":"IF(b<>0,(lambda1-a)/b,IF(AND(b=0,cc<>0),1,IF(AND(b=0,cc=0),0,\"\")))","v":{"v":1.6803940504246953,"d":"1.680"}},{"$f":"IF(b<>0,(lambda2-a)/b,IF(AND(b=0,cc<>0),1,IF(AND(b=0,cc=0),0,\"\")))","v":{"v":-2.3803940504246954,"d":"-2.380"}},{"$nulls":10},{"$f":"IF(bx<>0,(lambdax1-ax)/bx,IF(AND(bx=0,cx<>0),1,IF(AND(bx=0,cx=0),0,\"\")))","v":{"v":1.8556685669923318,"d":"1.856"}},{"$f":"IF(bx<>0,(lambdax2-ax)/bx,IF(AND(bx=0,cx<>0),1,IF(AND(bx=0,cx=0),0,\"\")))","v":{"v":-2.2956685669923322,"d":"-2.296"}},{"$nulls":8},{"$f":"IF(b0<>0,(lambda03-a0)/b0,IF(AND(b0=0,c0<>0),1,IF(AND(b0=0,c0=0),0,\"\")))","v":{"$error":"#VALUE!"}},{"$f":"IF(b0<>0,(lambda04-a0)/b0,IF(AND(b0=0,c0<>0),1,IF(AND(b0=0,c0=0),0,\"\")))","v":{"$error":"#VALUE!"}},{"$nulls":5},"x","y"]
[{"$nulls":22},"D < 0","lambda5 =",{"$f":"(a0+d0)/2","v":2.35},{"$f":"IF(Disc01<0,SQRT(-Disc01)/2,\"\")","v":""},"i",{"$nulls":3},{"$f":"IF(Disc00>0,(lambda001+lambda002)/2-3,IF(Disc00=0,lambda003-2,(a00+b00)/2-2))","v":6.645},{"$f":"AE13^2-(a00+d00)*AE13+(a00*d00-b00*c00)","v":-82.06602499999998}]
[{"$nulls":23},"lambda6 =",{"$f":"(a0+d0)/2","v":2.35},{"$f":"IF(Disc01<0,-SQRT(-Disc01)/2,\"\")","v":""},"i",{"$nulls":3},{"$f":"IF(Disc00>0,(lambda002+lambda001)/2+3,IF(Disc00=0,lambda003+2,(a00+b00)/2+2))","v":12.645},-82.06602500000001]
[{"$nulls":22},"eigenvects:",{"$f":"IF(b0<>0,1,IF(AND(b0=0,c0<>0),(alpha05-d0)/c0,IF(AND(b0=0,c0=0),0)))","v":1},{"$f":"IF(b0<>0,0,IF(AND(b0=0,c0<>0),beta05/c0,IF(AND(b0=0,c0=0),0)))","v":0},"i",{"$nulls":4},{"$f":"(AE13+AE14)/2","v":9.645},-91.066025]
["D = 0",{"$nulls":11},"D = 0",{"$nulls":10},{"$f":"IF(b0<>0,(alpha5-a0)/b0,IF(AND(b0=0,c0<>0),1,IF(AND(b0=0,c0=0),0)))","v":-0.3500000000000001},{"$f":"IF(b0<>0,beta05/b0,IF(AND(b0=0,c0<>0),0,IF(AND(b0=0,c0=0),0)))","v":{"$error":"#VALUE!"}},"i",{"$nulls":4},{"$f":"IF(Disc00>0,lambda001,IF(Disc00=0,lambda003,\"\"))","v":19.18785203699607},1.4210854715202004E-14]
["lambda3 =",{"$f":"IF(Disc=0,((a+d)/2),\"\")","v":""},{"$nulls":4},"x0 =",{"$f":"IF(Disc>0,B11,IF(Disc=0,B21,B31))","v":1},null,{"$f":"IF(Disc>0,C11,IF(Disc=0,C21,C31))","v":1},null,null,"lambdax3 =",{"$f":"IF(Discx=0,((ax+dx)/2),\"\")","v":""},{"$nulls":4},"x0 =",{"$f":"IF(Disc>0,N11,IF(Disc=0,N21,N31))","v":1},null,{"$f":"IF(Disc>0,O11,IF(Disc=0,O21,O31))","v":1},{"$nulls":8},{"$f":"IF(Disc00>0,lambda001,IF(Disc00=0,lambda004,\"\"))","v":19.18785203699607},1.4210854715202004E-14]
["lambda4 =",{"$f":"IF(Disc=0,((a+d)/2),\"\")","v":""},{"$nulls":4},"y0 =",{"$f":"IF(Disc>0,B12,IF(Disc=0,B22,B32))","v":{"v":1.6803940504246953,"d":"1.68"}},null,{"$f":"IF(Disc>0,C12,IF(Disc=0,C22,C32))","v":{"v":-2.3803940504246954,"d":"-2.38"}},null,null,"lambdax4 =",{"$f":"IF(Discx=0,((ax+dx)/2),\"\")","v":""},{"$nulls":4},"y0 =",{"$f":"IF(Disc>0,N12,IF(Disc=0,N22,N32))","v":{"v":1.8556685669923318,"d":"1.86"}},null,{"$f":"IF(Disc>0,O12,IF(Disc=0,O22,O32))","v":{"v":-2.2956685669923322,"d":"-2.30"}},{"$nulls":8},{"$f":"IF(Disc00>0,lambda002,IF(Disc00=0,lambda004,\"\"))","v":0.10214796300393125},1.2434497875801753E-14]
[{"$nulls":23},"A2 =",{"$f":"MMULT(_xlfn.ANCHORARRAY(Y1),_xlfn.ANCHORARRAY(Y1))","v":11.290000000000001},4.7]
["eigenvectors:",{"$nulls":11},"eigenvectors:",{"$nulls":11},18.8,8]
[null,{"$f":"IF(b<>0,1,IF(AND(b=0,cc<>0),(lambda3-d)/cc,IF(AND(b=0,cc=0),0)))","v":1},{"$f":"IF(b<>0,1,IF(AND(b=0,cc<>0),(lambda4-d)/cc,IF(AND(b=0,cc=0),0)))","v":1},{"$nulls":9},"???",{"$f":"IF(bx<>0,1,IF(AND(bx=0,cx<>0),(lambdax3-dx)/cx,IF(AND(bx=0,cx=0),0)))","v":1},{"$f":"IF(bx<>0,1,IF(AND(bx=0,cx<>0),(lambdax4-dx)/cx,IF(AND(bx=0,cx=0),0)))","v":1},{"$nulls":8},"Disc =",{"$f":"(a00+d00)^2-4*(a00*d00-b00*c00)","v":364.2641},{"$nulls":5},"x","y"]
[null,{"$f":"IF(b<>0,(lambda3-a)/b,IF(AND(b=0,cc<>0),1,IF(AND(b=0,cc=0),0,\"\")))","v":{"$error":"#VALUE!"}},{"$f":"IF(b<>0,(lambda4-a)/b,IF(AND(b=0,cc<>0),1,IF(AND(b=0,cc=0),0,\"\")))","v":{"$error":"#VALUE!"}},{"$nulls":10},{"$f":"IF(bx<>0,(lambdax3-ax)/bx,IF(AND(bx=0,cx<>0),1,IF(AND(bx=0,cx=0),0,\"\")))","v":{"$error":"#VALUE!"}},{"$f":"IF(bx<>0,(lambdax4-ax)/bx,IF(AND(bx=0,cx<>0),1,IF(AND(bx=0,cx=0),0,\"\")))","v":{"$error":"#VALUE!"}},null,null,{"$f":"lambdax3","v":""},{"$nulls":12},{"$f":"IF(Disc000>0,(lambda0001+lambda0002)/2-3,IF(Disc000=0,lambda0003-2,(a000+b000)/2-2))","v":39.0415},{"$f":"AE22^2-(a000+d000)*AE22+(a000*d000-b000*c000)","v":-1755.7437222500005}]
[{"$nulls":17},{"$f":"lambdax3","v":""},{"$nulls":4},"D > 0","lambda1 =",{"$f":"IF(Disc00>0,((a00+d00+SQRT(Disc00))/2),\"\")","v":{"v":19.18785203699607,"d":"19.188"}},{"$nulls":5},{"$f":"IF(Disc000>0,(lambda0002+lambda0001)/2+3,IF(Disc000=0,lambda0003+2,(a000+b000)/2+2))","v":45.0415},-1755.74372225]
[{"$nulls":23},"lambda2 =",{"$f":"IF(Disc00>0,((a00+d00-SQRT(Disc00))/2),\"\")","v":{"v":0.10214796300393125,"d":"0.102"}},{"$nulls":5},{"$f":"(AE22+AE23)/2","v":42.0415},-1764.7437222500002]
[{"$nulls":22},"eigenvects:",{"$f":"IF(b00<>0,1,IF(AND(b00=0,c00<>0),(lambda001-d00)/c00,IF(AND(b00=0,c00=0),0,\"\")))","v":1},{"$f":"IF(b00<>0,1,IF(AND(b00=0,c00<>0),(lambda002-d)/c00,IF(AND(b00=0,c00=0),0,\"\")))","v":1},{"$nulls":5},{"$f":"IF(Disc000>0,lambda0001,IF(Disc000=0,lambda0003,\"\"))","v":84.05035290328695},0]
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[]
[{"$nulls":22},"D > 0","lambda1 =",{"$f":"IF(Disc000>0,((a000+d000+SQRT(Disc000))/2),\"\")","v":{"v":84.05035290328695,"d":"84.050"}}]
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[]
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