the cpoy and paste of matrix will change the rank?

i use the following codes to get the matri of A ,

and the rank of A is 10,but when i copy the A and recalculate the rank of a,it runs out the 1,why?

codes are as this

omega=1.795200;

m0=7.234737;
m1=1.816835;
m2=5.349043;
m3=8.732933;
m4=12.024736;
m5=15.265388;
m6=18.476340;
m7=21.668910;
m8=24.849430;
m9=28.021668;
m10=31.187986;

syms a0 a1 a2   a3 a4  a5 a6 a7 a8 a9 alpha0 alpha1 alpha2 alpha3 alpha4 alpha5 alpha6 alpha7 alpha8 alpha9; 


h=22;

hgang=22;

agang=2.2497;

dgang=  21.1940;

a=agang/hgang;

d=dgang/hgang;


 
equation0=alpha0 == (2*a1*sin(d*m1)*besselk(1, a*m1))/(d*m1*(sin(2*m1)/(4*m1) + 1/2)^(1/2)) + (2*a2*sin(d*m2)*besselk(1, a*m2))/(d*m2*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) + (2*a3*sin(d*m3)*besselk(1, a*m3))/(d*m3*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) + (2*a4*sin(d*m4)*besselk(1, a*m4))/(d*m4*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) + (2*a0*sinh(d*m0)*(besselj(1, a*m0) + bessely(1, a*m0)*i))/(d*m0*(sinh(2*m0)/(4*m0) + 1/2)^(1/2));

equation1=alpha1 == (2*a1*d*m1*sin(d*m1)*besselk(1, a*m1))/((sin(2*m1)/(4*m1) + 1/2)^(1/2)*(pi^2 - d^2*m1^2)) + (2*a2*d*m2*sin(d*m2)*besselk(1, a*m2))/((sin(2*m2)/(4*m2) + 1/2)^(1/2)*(pi^2 - d^2*m2^2)) + (2*a3*d*m3*sin(d*m3)*besselk(1, a*m3))/((sin(2*m3)/(4*m3) + 1/2)^(1/2)*(pi^2 - d^2*m3^2)) + (2*a4*d*m4*sin(d*m4)*besselk(1, a*m4))/((sin(2*m4)/(4*m4) + 1/2)^(1/2)*(pi^2 - d^2*m4^2)) - (2*a0*d*m0*sinh(d*m0)*(besselj(1, a*m0) + bessely(1, a*m0)*i))/((sinh(2*m0)/(4*m0) + 1/2)^(1/2)*(pi^2 + d^2*m0^2));

equation2=alpha2 == (2*a0*d*m0*sinh(d*m0)*(besselj(1, a*m0) + bessely(1, a*m0)*i))/((4*pi^2 + d^2*m0^2)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)) - (2*a2*d*m2*sin(d*m2)*besselk(1, a*m2))/((4*pi^2 - d^2*m2^2)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) - (2*a3*d*m3*sin(d*m3)*besselk(1, a*m3))/((4*pi^2 - d^2*m3^2)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) - (2*a4*d*m4*sin(d*m4)*besselk(1, a*m4))/((4*pi^2 - d^2*m4^2)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) - (2*a1*d*m1*sin(d*m1)*besselk(1, a*m1))/((4*pi^2 - d^2*m1^2)*(sin(2*m1)/(4*m1) + 1/2)^(1/2));

equation3=alpha3 == (2*a1*d*m1*sin(d*m1)*besselk(1, a*m1))/((9*pi^2 - d^2*m1^2)*(sin(2*m1)/(4*m1) + 1/2)^(1/2)) + (2*a2*d*m2*sin(d*m2)*besselk(1, a*m2))/((9*pi^2 - d^2*m2^2)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) + (2*a3*d*m3*sin(d*m3)*besselk(1, a*m3))/((9*pi^2 - d^2*m3^2)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) + (2*a4*d*m4*sin(d*m4)*besselk(1, a*m4))/((9*pi^2 - d^2*m4^2)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) - (2*a0*d*m0*sinh(d*m0)*(besselj(1, a*m0) + bessely(1, a*m0)*i))/((9*pi^2 + d^2*m0^2)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2));

equation4=alpha4 == (2*a0*d*m0*sinh(d*m0)*(besselj(1, a*m0) + bessely(1, a*m0)*i))/((16*pi^2 + d^2*m0^2)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)) - (2*a2*d*m2*sin(d*m2)*besselk(1, a*m2))/((16*pi^2 - d^2*m2^2)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) - (2*a3*d*m3*sin(d*m3)*besselk(1, a*m3))/((16*pi^2 - d^2*m3^2)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) - (2*a4*d*m4*sin(d*m4)*besselk(1, a*m4))/((16*pi^2 - d^2*m4^2)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) - (2*a1*d*m1*sin(d*m1)*besselk(1, a*m1))/((16*pi^2 - d^2*m1^2)*(sin(2*m1)/(4*m1) + 1/2)^(1/2));

equation5=a0 == ((sinh(d*m0) - sinh(m0))/(m0*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)) - (alpha0*sinh(d*m0))/(2*a*m0*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)) + (pi*alpha1*d*m0*sinh(d*m0)*(besseli(2, a*m1) + besseli(1, a*m1)/(a*m1)))/(besseli(1, a*m1)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)*(pi^2 + d^2*m0^2)) - (2*pi*alpha2*d*m0*sinh(d*m0)*(besseli(2, a*m2) + besseli(1, a*m2)/(a*m2)))/((4*pi^2 + d^2*m0^2)*besseli(1, a*m2)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)) + (3*pi*alpha3*d*m0*sinh(d*m0)*(besseli(2, a*m3) + besseli(1, a*m3)/(a*m3)))/((9*pi^2 + d^2*m0^2)*besseli(1, a*m3)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)) - (4*pi*alpha4*d*m0*sinh(d*m0)*(besseli(2, a*m4) + besseli(1, a*m4)/(a*m4)))/((16*pi^2 + d^2*m0^2)*besseli(1, a*m4)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)))/(m0*(besselj(2, a*m0) + bessely(2, a*m0)*i - besselj(1, a*m0)/(a*m0) + (bessely(1, a*m0)*(-i))/(a*m0)));

equation6=a1 == ((sin(d*m1) - sin(m1))/(m1*(sin(2*m1)/(4*m1) + 1/2)^(1/2)) - (alpha0*sin(d*m1))/(2*a*m1*(sin(2*m1)/(4*m1) + 1/2)^(1/2)) - (pi*alpha1*d*m1*sin(d*m1)*(besseli(2, a*m1) + besseli(1, a*m1)/(a*m1)))/(besseli(1, a*m1)*(sin(2*m1)/(4*m1) + 1/2)^(1/2)*(pi^2 - d^2*m1^2)) + (2*pi*alpha2*d*m1*sin(d*m1)*(besseli(2, a*m2) + besseli(1, a*m2)/(a*m2)))/((4*pi^2 - d^2*m1^2)*besseli(1, a*m2)*(sin(2*m1)/(4*m1) + 1/2)^(1/2)) - (3*pi*alpha3*d*m1*sin(d*m1)*(besseli(2, a*m3) + besseli(1, a*m3)/(a*m3)))/((9*pi^2 - d^2*m1^2)*besseli(1, a*m3)*(sin(2*m1)/(4*m1) + 1/2)^(1/2)) + (4*pi*alpha4*d*m1*sin(d*m1)*(besseli(2, a*m4) + besseli(1, a*m4)/(a*m4)))/((16*pi^2 - d^2*m1^2)*besseli(1, a*m4)*(sin(2*m1)/(4*m1) + 1/2)^(1/2)))/(m1*(besselk(2, a*m1) - besselk(1, a*m1)/(a*m1)));

equation7=a2 == ((sin(d*m2) - sin(m2))/(m2*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) - (alpha0*sin(d*m2))/(2*a*m2*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) - (pi*alpha1*d*m2*sin(d*m2)*(besseli(2, a*m1) + besseli(1, a*m1)/(a*m1)))/(besseli(1, a*m1)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)*(pi^2 - d^2*m2^2)) + (2*pi*alpha2*d*m2*sin(d*m2)*(besseli(2, a*m2) + besseli(1, a*m2)/(a*m2)))/((4*pi^2 - d^2*m2^2)*besseli(1, a*m2)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) - (3*pi*alpha3*d*m2*sin(d*m2)*(besseli(2, a*m3) + besseli(1, a*m3)/(a*m3)))/((9*pi^2 - d^2*m2^2)*besseli(1, a*m3)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) + (4*pi*alpha4*d*m2*sin(d*m2)*(besseli(2, a*m4) + besseli(1, a*m4)/(a*m4)))/((16*pi^2 - d^2*m2^2)*besseli(1, a*m4)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)))/(m2*(besselk(2, a*m2) - besselk(1, a*m2)/(a*m2)));

equation8=a3 == ((sin(d*m3) - sin(m3))/(m3*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) - (alpha0*sin(d*m3))/(2*a*m3*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) - (pi*alpha1*d*m3*sin(d*m3)*(besseli(2, a*m1) + besseli(1, a*m1)/(a*m1)))/(besseli(1, a*m1)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)*(pi^2 - d^2*m3^2)) + (2*pi*alpha2*d*m3*sin(d*m3)*(besseli(2, a*m2) + besseli(1, a*m2)/(a*m2)))/((4*pi^2 - d^2*m3^2)*besseli(1, a*m2)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) - (3*pi*alpha3*d*m3*sin(d*m3)*(besseli(2, a*m3) + besseli(1, a*m3)/(a*m3)))/((9*pi^2 - d^2*m3^2)*besseli(1, a*m3)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) + (4*pi*alpha4*d*m3*sin(d*m3)*(besseli(2, a*m4) + besseli(1, a*m4)/(a*m4)))/((16*pi^2 - d^2*m3^2)*besseli(1, a*m4)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)))/(m3*(besselk(2, a*m3) - besselk(1, a*m3)/(a*m3)));

equation9=a4 == ((sin(d*m4) - sin(m4))/(m4*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) - (alpha0*sin(d*m4))/(2*a*m4*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) - (pi*alpha1*d*m4*sin(d*m4)*(besseli(2, a*m1) + besseli(1, a*m1)/(a*m1)))/(besseli(1, a*m1)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)*(pi^2 - d^2*m4^2)) + (2*pi*alpha2*d*m4*sin(d*m4)*(besseli(2, a*m2) + besseli(1, a*m2)/(a*m2)))/((4*pi^2 - d^2*m4^2)*besseli(1, a*m2)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) - (3*pi*alpha3*d*m4*sin(d*m4)*(besseli(2, a*m3) + besseli(1, a*m3)/(a*m3)))/((9*pi^2 - d^2*m4^2)*besseli(1, a*m3)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) + (4*pi*alpha4*d*m4*sin(d*m4)*(besseli(2, a*m4) + besseli(1, a*m4)/(a*m4)))/((16*pi^2 - d^2*m4^2)*besseli(1, a*m4)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)))/(m4*(besselk(2, a*m4) - besselk(1, a*m4)/(a*m4)));



[A, b] = equationsToMatrix([equation0, equation1, equation2,equation3,equation4,equation5,equation6, equation7, equation8,equation9], [a0, a1, a2 ,a3, a4 , alpha0 ,alpha1 ,alpha2 ,alpha3 ,alpha4]);

and the rank( A) is 10，

but when i copt the matrix A and calculate the rank of A,it get out the 1,why?

A =...
 ...
[ - 14547186153589216317120976241525/50328715046552884871112366751744 - (11067376412618097943977112897985*i)/12582178761638221217778091687936, 10045566870117554386053609892956350802656748920209/41815826252695678451841357052067724394233067470848 + (38212912330913546333933271927439582529157556319393*i)/52269782815869598064801696315084655492791334338560, - 10045566870117554386053609892956350802656748920209/62999744934768208151332428452243698833687383638016 - (38212912330913546333933271927439582529157556319393*i)/78749681168460260189165535565304623542109229547520, 1435080981445364912293372841850907257522392702887/14043753724507969458302419437250801878643960184832 + (126953197112669589149279973180862400429094871493*i)/408248654782208414485535448757290752286161633280, - 10045566870117554386053609892956350802656748920209/147735419663058348367521255509111564279802006339584 - (38212912330913546333933271927439582529157556319393*i)/184669274578822935459401569386389455349752507924480,                                                                                                                                                                                                                                                                                        1,                                                                                                            0,                                                                                                           0,                                                                                                              0,                                                                                                              0]
[                                                                               -296349324225323643384292415563/33639933814238323125075836928,                                                                                                              -8129216622937086741211209072144761934456574976889/2050170152471208841565167065708702466276170137600,                                                                                                                 2709738874312362247070403024048253978152191658963/3656343846794279512703106531818576750862758051840,                                                                                                        -8129216622937086741211209072144761934456574976889/25833800520235551318614639869755705905233233182720,                                                                                                                  8129216622937086741211209072144761934456574976889/46644477092029352456507624838042939650496381583360,                                                                                                                                                                                                                                                                                        0,                                                                                                            1,                                                                                                           0,                                                                                                              0,                                                                                                              0]
[                                                                           13522588663836194993980917796725/17631211176203321124428655362048,                                                                                                              -3452062385466744459959884618289904701226396322167/2828023964629027589888871736420775202197262565376,                                                                                                                -3452062385466744459959884618289904701226396322167/2190653302455404848074155102501142206346409541632,                                                                                                               3497530279094979189422375499787137488577909141/10694139224177094924916120683256758406712655872,                                                                                                                 -3452062385466744459959884618289904701226396322167/22265362370793135938565296299199059821039682846720,                                                                                                                                                                                                                                                                                        0,                                                                                                            0,                                                                                                           1,                                                                                                              0,                                                                                                              0]
[                                                                            -1559505055508291766081383701859/7325920830152149842646987177984,                                                                                                            40622780053623480499550084142509355499320739442737/164219357544379496541994966008429797107988168704000,                                                                                                              -40622780053623480499550084142509355499320739442737/84389506161396227241690508417791717252379181056000,                                                                                                       -40622780053623480499550084142509355499320739442737/48660246143575895230895222155200124072170094592000,                                                                                                                40622780053623480499550084142509355499320739442737/234929899370536863903683924455284018633792028672000,                                                                                                                                                                                                                                                                                        0,                                                                                                            0,                                                                                                           0,                                                                                                              1,                                                                                                              0]
[                                                                              145838977485677560441873737143/1698523909862782370727849885696,                                                                                                             -5230845433647037967648954968353982549043744360977/56440789552437185617852909254801587025922228224000,                                                                                                                5230845433647037967648954968353982549043744360977/42998930339625539910443664973947233039634399232000,                                                                                                       -15692536300941113902946864905061947647131233082931/61787494954818384222206472262308220955596423168000,                                                                                                                 -5230845433647037967648954968353982549043744360977/10768506511621049891271613404731891137384546304000,                                                                                                                                                                                                                                                                                        0,                                                                                                            0,                                                                                                           0,                                                                                                              0,                                                                                                              1]
[                                                                                                                                           1,                                                                                                                                                                                                                 0,                                                                                                                                                                                                                   0,                                                                                                                                                                                                            0,                                                                                                                                                                                                                     0,                                                                                              - 19215206817301820201937645933374036236042240/312547901147939322718983515605629276594765441 - (6768960512610041956879930131725477048483840*i)/34727544571993258079887057289514364066085049,                                               311822173964424859837463003136/4182101353199550343907676068867,                                               -32758260305221388623706849280/507157969485712932497800171043,                                               1074690585838732255773514006528/17714189487407758346830959806935,                                              -4217261388076414059818656989184/66954719808397516799363753130003]
[                                                                                                                                           0,                                                                                                                                                                                                                 1,                                                                                                                                                                                                                   0,                                                                                                                                                                                                            0,                                                                                                                                                                                                                     0,              pi*(144059190304636750863338087835057724199218497968354586295107797/2446196724593975793015010125110610682116089828997016170381967360 + (114182725422888477127721339727956710052886909222178814898704217*i)/611549181148493948253752531277652670529022457249254042595491840),   (6373987289404659872595495466573082135095775602773*pi)/164792814120371931452773292525785545285435769487360,         (1866902816615813251928194064653293767800945463*pi)/15757081295075041034691086819326489962244734976,     -(5155874420064302306074166750515442461692706494419*pi)/63455452499346924584569286752621495539451559936000,    (13929477699905966940085900270602586319233983248593*pi)/177769867492423376011634069286936746537615622144000]
[                                                                                                                                           0,                                                                                                                                                                                                                 0,                                                                                                                                                                                                                   1,                                                                                                                                                                                                            0,                                                                                                                                                                                                                     0,      pi*(- 9464989338576010663641040107852943348161756038187711210495130309/334786751097527661443341868614963494529183647077160847702082191360 - (7502043267713731825867055861261399052044045856414964256985623049*i)/83696687774381915360835467153740873632295911769290211925520547840),    -(359647875850784208474461056105259465318300911317*pi)/68783230572620007852236149854508416087444800143360,   (1491280192527213932355605863890205848162693411339*pi)/13480422184672866986284910016233667451715615457280,  (124803259942280896090021788864765346762906143405079*pi)/1091330405129737807036538042638191289801382035456000, -(337177379549110141775193947667076009201637100775613*pi)/4532584749598244496474862383717467767011759095808000]
[                                                                                                                                           0,                                                                                                                                                                                                                 0,                                                                                                                                                                                                                   0,                                                                                                                                                                                                            1,                                                                                                                                                                                                                     0,     pi*(15646162213207041479098502059587806306703894195417071959648052421/846030272049793392224269349451262918787793849710511133560980635648 + (12401301438212310647408758642027689291628507785318486157217897481*i)/211507568012448348056067337362815729696948462427627783390245158912),  (9936948082441062162015806310171888189711714626463*pi)/4384959577658516572786543633942597099272009024536576, -(5282508055207274506629691783028660420071424958195*pi)/225403800818533532313654072351950733940600273371136,       (1403447897422781626477159087055068774733031825133*pi)/6932660975349988900289667134781792796272492544000,      (3791654838227565187249719560444373602394912477551*pi)/23918033291036642220972615444403915409124753408000]
[                                                                                                                                           0,                                                                                                                                                                                                                 0,                                                                                                                                                                                                                   0,                                                                                                                                                                                                            0,                                                                                                                                                                                                                     1, pi*(- 96850311095890057474600622066292417300526063865839919947163193497/7149692360286571710357404675869855841740536003322368113589050408960 - (76764505309227784444080920316982007181254868855370344850327591917*i)/1787423090071642927589351168967463960435134000830592028397262602240), -(2278150693781893501965631214056678334135659812033*pi)/1648966467348438933504394877733030378029235977584640,    (726642584284646511025523077002058332515134227547*pi)/59417389819028630748491999550292796349374886051840, -(182435260226832054739117358996020969042983792915701*pi)/3952585887016323880807243202099537361549452640256000,  (492880097916447269765789180295077779652228882159047*pi)/1476790770714909314421347029553103020708755144704000]

and also when i use the

fprintf('A=%s;',char(A))

to print A and get that the rank of A is 1,how queer.

Answer 1

[Walter Roberson]

When you copy and paste

 - 14547186153589216317120976241525/50328715046552884871112366751744 - (11067376412618097943977112897985*i)/12582178761638221217778091687936

then those are evaluated as double precision numbers.

14547186153589216317120976241525 approximates as 14547186153589216981797711118336

50328715046552884871112366751744 approximates as itself

11067376412618097943977112897985 approximates as 11067376412618098658996614332416

12582178761638221217778091687936 approximates as itself

With the numerators in the real and imaginary parts both being approximated, it is not uncommon that the rank of the matrix will change.

Remember, just because you can get the Symbolic Toolbox to output a long integer does not mean that regular MATLAB can work with numbers those long without losing precision.

  - sym('14547186153589216317120976241525')/sym('50328715046552884871112366751744') - (sym('11067376412618097943977112897985')*i)/sym('12582178761638221217778091687936')

[dcydhb dcydhb]

But I copied all the results, and the result was that the fractions and sym(‘pi’), not the decimals ,didn't have truncation characters, so how could there be an approximation?

[Walter Roberson]

A =...
 ...
[ - 14547186153589216317120976241525/50328715046552884871112366751744 - (11067376412618097943977112897985*i)/12582178761638221217778091687936, 10045566870117554386053609892956350802656748920209/41815826252695678451841357052067724394233067470848 + (38212912330913546333933271927439582529157556319393*i)/52269782815869598064801696315084655492791334338560, - 10045566870117554386053609892956350802656748920209/62999744934768208151332428452243698833687383638016 - (38212912330913546333933271927439582529157556319393*i)/78749681168460260189165535565304623542109229547520, 1435080981445364912293372841850907257522392702887/14043753724507969458302419437250801878643960184832 + (126953197112669589149279973180862400429094871493*i)/408248654782208414485535448757290752286161633280, - 10045566870117554386053609892956350802656748920209/147735419663058348367521255509111564279802006339584 - (38212912330913546333933271927439582529157556319393*i)/184669274578822935459401569386389455349752507924480,                                                                                                                                                                                                                                                                                        1,                                                                                                            0,                                                                                                           0,                                                                                                              0,                                                                                                              0]
[                                                                               -296349324225323643384292415563/33639933814238323125075836928,                                                                                                              -8129216622937086741211209072144761934456574976889/2050170152471208841565167065708702466276170137600,                                                                                                                 2709738874312362247070403024048253978152191658963/3656343846794279512703106531818576750862758051840,                                                                                                        -8129216622937086741211209072144761934456574976889/25833800520235551318614639869755705905233233182720,                                                                                                                  8129216622937086741211209072144761934456574976889/46644477092029352456507624838042939650496381583360,                                                                                                                                                                                                                                                                                        0,                                                                                                            1,                                                                                                           0,                                                                                                              0,                                                                                                              0]
[                                                                           13522588663836194993980917796725/17631211176203321124428655362048,                                                                                                              -3452062385466744459959884618289904701226396322167/2828023964629027589888871736420775202197262565376,                                                                                                                -3452062385466744459959884618289904701226396322167/2190653302455404848074155102501142206346409541632,                                                                                                               3497530279094979189422375499787137488577909141/10694139224177094924916120683256758406712655872,                                                                                                                 -3452062385466744459959884618289904701226396322167/22265362370793135938565296299199059821039682846720,                                                                                                                                                                                                                                                                                        0,                                                                                                            0,                                                                                                           1,                                                                                                              0,                                                                                                              0]
[                                                                            -1559505055508291766081383701859/7325920830152149842646987177984,                                                                                                            40622780053623480499550084142509355499320739442737/164219357544379496541994966008429797107988168704000,                                                                                                              -40622780053623480499550084142509355499320739442737/84389506161396227241690508417791717252379181056000,                                                                                                       -40622780053623480499550084142509355499320739442737/48660246143575895230895222155200124072170094592000,                                                                                                                40622780053623480499550084142509355499320739442737/234929899370536863903683924455284018633792028672000,                                                                                                                                                                                                                                                                                        0,                                                                                                            0,                                                                                                           0,                                                                                                              1,                                                                                                              0]
[                                                                              145838977485677560441873737143/1698523909862782370727849885696,                                                                                                             -5230845433647037967648954968353982549043744360977/56440789552437185617852909254801587025922228224000,                                                                                                                5230845433647037967648954968353982549043744360977/42998930339625539910443664973947233039634399232000,                                                                                                       -15692536300941113902946864905061947647131233082931/61787494954818384222206472262308220955596423168000,                                                                                                                 -5230845433647037967648954968353982549043744360977/10768506511621049891271613404731891137384546304000,                                                                                                                                                                                                                                                                                        0,                                                                                                            0,                                                                                                           0,                                                                                                              0,                                                                                                              1]
[                                                                                                                                           1,                                                                                                                                                                                                                 0,                                                                                                                                                                                                                   0,                                                                                                                                                                                                            0,                                                                                                                                                                                                                     0,                                                                                              - 19215206817301820201937645933374036236042240/312547901147939322718983515605629276594765441 - (6768960512610041956879930131725477048483840*i)/34727544571993258079887057289514364066085049,                                               311822173964424859837463003136/4182101353199550343907676068867,                                               -32758260305221388623706849280/507157969485712932497800171043,                                               1074690585838732255773514006528/17714189487407758346830959806935,                                              -4217261388076414059818656989184/66954719808397516799363753130003]
[                                                                                                                                           0,                                                                                                                                                                                                                 1,                                                                                                                                                                                                                   0,                                                                                                                                                                                                            0,                                                                                                                                                                                                                     0,              pi*(144059190304636750863338087835057724199218497968354586295107797/2446196724593975793015010125110610682116089828997016170381967360 + (114182725422888477127721339727956710052886909222178814898704217*i)/611549181148493948253752531277652670529022457249254042595491840),   (6373987289404659872595495466573082135095775602773*pi)/164792814120371931452773292525785545285435769487360,         (1866902816615813251928194064653293767800945463*pi)/15757081295075041034691086819326489962244734976,     -(5155874420064302306074166750515442461692706494419*pi)/63455452499346924584569286752621495539451559936000,    (13929477699905966940085900270602586319233983248593*pi)/177769867492423376011634069286936746537615622144000]
[                                                                                                                                           0,                                                                                                                                                                                                                 0,                                                                                                                                                                                                                   1,                                                                                                                                                                                                            0,                                                                                                                                                                                                                     0,      pi*(- 9464989338576010663641040107852943348161756038187711210495130309/334786751097527661443341868614963494529183647077160847702082191360 - (7502043267713731825867055861261399052044045856414964256985623049*i)/83696687774381915360835467153740873632295911769290211925520547840),    -(359647875850784208474461056105259465318300911317*pi)/68783230572620007852236149854508416087444800143360,   (1491280192527213932355605863890205848162693411339*pi)/13480422184672866986284910016233667451715615457280,  (124803259942280896090021788864765346762906143405079*pi)/1091330405129737807036538042638191289801382035456000, -(337177379549110141775193947667076009201637100775613*pi)/4532584749598244496474862383717467767011759095808000]
[                                                                                                                                           0,                                                                                                                                                                                                                 0,                                                                                                                                                                                                                   0,                                                                                                                                                                                                            1,                                                                                                                                                                                                                     0,     pi*(15646162213207041479098502059587806306703894195417071959648052421/846030272049793392224269349451262918787793849710511133560980635648 + (12401301438212310647408758642027689291628507785318486157217897481*i)/211507568012448348056067337362815729696948462427627783390245158912),  (9936948082441062162015806310171888189711714626463*pi)/4384959577658516572786543633942597099272009024536576, -(5282508055207274506629691783028660420071424958195*pi)/225403800818533532313654072351950733940600273371136,       (1403447897422781626477159087055068774733031825133*pi)/6932660975349988900289667134781792796272492544000,      (3791654838227565187249719560444373602394912477551*pi)/23918033291036642220972615444403915409124753408000]
[                                                                                                                                           0,                                                                                                                                                                                                                 0,                                                                                                                                                                                                                   0,                                                                                                                                                                                                            0,                                                                                                                                                                                                                     1, pi*(- 96850311095890057474600622066292417300526063865839919947163193497/7149692360286571710357404675869855841740536003322368113589050408960 - (76764505309227784444080920316982007181254868855370344850327591917*i)/1787423090071642927589351168967463960435134000830592028397262602240), -(2278150693781893501965631214056678334135659812033*pi)/1648966467348438933504394877733030378029235977584640,    (726642584284646511025523077002058332515134227547*pi)/59417389819028630748491999550292796349374886051840, -(182435260226832054739117358996020969042983792915701*pi)/3952585887016323880807243202099537361549452640256000,  (492880097916447269765789180295077779652228882159047*pi)/1476790770714909314421347029553103020708755144704000]

means the same as

A = [ - 14547186153589216317120976241525/50328715046552884871112366751744 - (11067376412618097943977112897985*i)/12582178761638221217778091687936, 10045566870117554386053609892956350802656748920209/41815826252695678451841357052067724394233067470848 + (38212912330913546333933271927439582529157556319393*i)/52269782815869598064801696315084655492791334338560, - 10045566870117554386053609892956350802656748920209/62999744934768208151332428452243698833687383638016 - (38212912330913546333933271927439582529157556319393*i)/78749681168460260189165535565304623542109229547520, 1435080981445364912293372841850907257522392702887/14043753724507969458302419437250801878643960184832 + (126953197112669589149279973180862400429094871493*i)/408248654782208414485535448757290752286161633280, - 10045566870117554386053609892956350802656748920209/147735419663058348367521255509111564279802006339584 - (38212912330913546333933271927439582529157556319393*i)/184669274578822935459401569386389455349752507924480,                                                                                                                                                                                                                                                                                        1,                                                                                                            0,                                                                                                           0,                                                                                                              0,                                                                                                              0];
disp(A)

disp([                                                                               -296349324225323643384292415563/33639933814238323125075836928,                                                                                                              -8129216622937086741211209072144761934456574976889/2050170152471208841565167065708702466276170137600,                                                                                                                 2709738874312362247070403024048253978152191658963/3656343846794279512703106531818576750862758051840,                                                                                                        -8129216622937086741211209072144761934456574976889/25833800520235551318614639869755705905233233182720,                                                                                                                  8129216622937086741211209072144761934456574976889/46644477092029352456507624838042939650496381583360,                                                                                                                                                                                                                                                                                        0,                                                                                                            1,                                                                                                           0,                                                                                                              0,                                                                                                              0])

disp([                                                                           13522588663836194993980917796725/17631211176203321124428655362048,                                                                                                              -3452062385466744459959884618289904701226396322167/2828023964629027589888871736420775202197262565376,                                                                                                                -3452062385466744459959884618289904701226396322167/2190653302455404848074155102501142206346409541632,                                                                                                               3497530279094979189422375499787137488577909141/10694139224177094924916120683256758406712655872,                                                                                                                 -3452062385466744459959884618289904701226396322167/22265362370793135938565296299199059821039682846720,                                                                                                                                                                                                                                                                                        0,                                                                                                            0,                                                                                                           1,                                                                                                              0,                                                                                                              0])

disp([                                                                            -1559505055508291766081383701859/7325920830152149842646987177984,                                                                                                            40622780053623480499550084142509355499320739442737/164219357544379496541994966008429797107988168704000,                                                                                                              -40622780053623480499550084142509355499320739442737/84389506161396227241690508417791717252379181056000,                                                                                                       -40622780053623480499550084142509355499320739442737/48660246143575895230895222155200124072170094592000,                                                                                                                40622780053623480499550084142509355499320739442737/234929899370536863903683924455284018633792028672000,                                                                                                                                                                                                                                                                                        0,                                                                                                            0,                                                                                                           0,                                                                                                              1,                                                                                                              0])

disp([                                                                              145838977485677560441873737143/1698523909862782370727849885696,                                                                                                             -5230845433647037967648954968353982549043744360977/56440789552437185617852909254801587025922228224000,                                                                                                                5230845433647037967648954968353982549043744360977/42998930339625539910443664973947233039634399232000,                                                                                                       -15692536300941113902946864905061947647131233082931/61787494954818384222206472262308220955596423168000,                                                                                                                 -5230845433647037967648954968353982549043744360977/10768506511621049891271613404731891137384546304000,                                                                                                                                                                                                                                                                                        0,                                                                                                            0,                                                                                                           0,                                                                                                              0,                                                                                                              1])

disp([                                                                                                                                           1,                                                                                                                                                                                                                 0,                                                                                                                                                                                                                   0,                                                                                                                                                                                                            0,                                                                                                                                                                                                                     0,                                                                                              - 19215206817301820201937645933374036236042240/312547901147939322718983515605629276594765441 - (6768960512610041956879930131725477048483840*i)/34727544571993258079887057289514364066085049,                                               311822173964424859837463003136/4182101353199550343907676068867,                                               -32758260305221388623706849280/507157969485712932497800171043,                                               1074690585838732255773514006528/17714189487407758346830959806935,                                              -4217261388076414059818656989184/66954719808397516799363753130003])

disp([                                                                                                                                           0,                                                                                                                                                                                                                 1,                                                                                                                                                                                                                   0,                                                                                                                                                                                                            0,                                                                                                                                                                                                                     0,              pi*(144059190304636750863338087835057724199218497968354586295107797/2446196724593975793015010125110610682116089828997016170381967360 + (114182725422888477127721339727956710052886909222178814898704217*i)/611549181148493948253752531277652670529022457249254042595491840),   (6373987289404659872595495466573082135095775602773*pi)/164792814120371931452773292525785545285435769487360,         (1866902816615813251928194064653293767800945463*pi)/15757081295075041034691086819326489962244734976,     -(5155874420064302306074166750515442461692706494419*pi)/63455452499346924584569286752621495539451559936000,    (13929477699905966940085900270602586319233983248593*pi)/177769867492423376011634069286936746537615622144000])

disp([                                                                                                                                           0,                                                                                                                                                                                                                 0,                                                                                                                                                                                                                   1,                                                                                                                                                                                                            0,                                                                                                                                                                                                                     0,      pi*(- 9464989338576010663641040107852943348161756038187711210495130309/334786751097527661443341868614963494529183647077160847702082191360 - (7502043267713731825867055861261399052044045856414964256985623049*i)/83696687774381915360835467153740873632295911769290211925520547840),    -(359647875850784208474461056105259465318300911317*pi)/68783230572620007852236149854508416087444800143360,   (1491280192527213932355605863890205848162693411339*pi)/13480422184672866986284910016233667451715615457280,  (124803259942280896090021788864765346762906143405079*pi)/1091330405129737807036538042638191289801382035456000, -(337177379549110141775193947667076009201637100775613*pi)/4532584749598244496474862383717467767011759095808000])
disp([                                                                                                                                           0,                                                                                                                                                                                                                 0,                                                                                                                                                                                                                   0,                                                                                                                                                                                                            1,                                                                                                                                                                                                                     0,     pi*(15646162213207041479098502059587806306703894195417071959648052421/846030272049793392224269349451262918787793849710511133560980635648 + (12401301438212310647408758642027689291628507785318486157217897481*i)/211507568012448348056067337362815729696948462427627783390245158912),  (9936948082441062162015806310171888189711714626463*pi)/4384959577658516572786543633942597099272009024536576, -(5282508055207274506629691783028660420071424958195*pi)/225403800818533532313654072351950733940600273371136,       (1403447897422781626477159087055068774733031825133*pi)/6932660975349988900289667134781792796272492544000,      (3791654838227565187249719560444373602394912477551*pi)/23918033291036642220972615444403915409124753408000])

disp([                                                                                                                                           0,                                                                                                                                                                                                                 0,                                                                                                                                                                                                                   0,                                                                                                                                                                                                            0,                                                                                                                                                                                                                     1, pi*(- 96850311095890057474600622066292417300526063865839919947163193497/7149692360286571710357404675869855841740536003322368113589050408960 - (76764505309227784444080920316982007181254868855370344850327591917*i)/1787423090071642927589351168967463960435134000830592028397262602240), -(2278150693781893501965631214056678334135659812033*pi)/1648966467348438933504394877733030378029235977584640,    (726642584284646511025523077002058332515134227547*pi)/59417389819028630748491999550292796349374886051840, -(182435260226832054739117358996020969042983792915701*pi)/3952585887016323880807243202099537361549452640256000,  (492880097916447269765789180295077779652228882159047*pi)/1476790770714909314421347029553103020708755144704000])

That is, having a bunch of [] on consecutive rows does not mean that the rows take part in the previous assignment. Not unless there are ;... on the ends of the lines.

[dcydhb dcydhb]

so you mean,

A =...
    ...
    [1 2 3]
[4 5 6]
[7 8 10];

rank(A )=1,

and when

A1 =...
    ...
    [1 2 3
4 5 6]
7 8 10];

now rank(A1)=3,

and also we can use

A2 =...
    ...
    [[1 2 3]
[4 5 6]
[7 8 10]];

and now rank(A2) =3,

so it is just not the approximatation but the assignment of the matrix？

[Walter Roberson]

A =...
    ...
    [1 2 3]

The assigment ends after that line.

[4 5 6]

There is no assignment happening, and there is no semi-colon, so a vector is built and the result is displayed.

[7 8 10];

There is no assignment happening, and there is a semi-colon on the end of the line, so a vector is built and the result is discarded.

Assignment continues automatically on to a new line under any of three different circumstances:

1. The active part of the line ends in ... When that happens, the rest of that line is discarded, and the result is as if the next line was there in place of the ... with the line break being ignored . [123 ... line-break 456] is the same as [123 456] not the same as [123; 456]
2. there is an active [ that has not been balanced by a ] . When that happens, unless the line ended with ... then the line is treated as-if it ended with ;... so [123 line-break 456] is treated the same as [123 ;... line-break 456] which is the same as [123; 456]
3. there is an active { that has not been balanced by a } . When that happens, unless the line ended with ... then the line is treated as-if it ended with ;... so {123 line-break 456} is treated the same as {123 ;... line-break 456} which is the same as {123; 456}