This is an example of what I think is a misdirection in the WRI approach, one that could probably be easily fixed. That is what I call the lack of "hierarchical depth", the ability to calculate things at all levels. This is partly a result of concentrating too much on high powered, top-down routines. If you want to check them with lower level routines it's all hidden and inaccessible. The more basic routines are either missing or not tailored to more basic usage. It's not that I don't think there should be high level "set-piece" routines for common applications, but just that users should have access to lower level approaches whenever possible. Anyway, I always go for "hierarchical depth" and "calculate everything".
In Presentations I have a Student's Linear Equations section that allows step by step manipulation of matrix structures. I never envisioned it for a research problem as discussed in the PDF. Nevertheless, I thought I would experiment. I stumbled into calculating the determinant by calculating the reciprocal of the determinate of the inverse. In a case that was giving discordant and incorrect results with the straight determinant I obtained consistent, and what looks like correct results. Here is the test case:
powersMatrix =
DiagonalMatrix[
10^# & /@ {123, 152, 185, 220, 397, 449, 503, 563, 979, 1059, 1143,
1229, 1319, 1412}];
basicMatrix = {{39, 73, -86, -75, 24, -67, -87, 9, -65, -98, 60, 85, 59, -65}, {14,
75, 84, 63, -26, -15, -41, 98, -57, 31, 72, -99, 10, -18}, {-47,
84, -92, -39, 56, -59, -40, -77, -37, -47, 50, 31, -95, -78}, {0,
64, -76, -23, -95, 23, -30, 23, -46, -3, -21, 79, 1, 91}, {29,
42, -97, -10, 60, -84, 47, 84, -79, -2, 78, 40, 67, 95}, {-30, 31,
23, -78, 28, 16, 3, -91, -13, -80, -60, -29,
15, -22}, {-4, -10, -74, -17, -89, -48,
59, -78, -77, -59, -48, -59, -20, 41}, {72, 68, 35, -83, -63,
55, -27, 86, -3, -19, 51, 91, -9, -20}, {9, 95, -39, 21, 64, -61,
23, 93, 38, -68, -95, -28, -61, 19}, {-67, 91, 44, -97, -43, 93, 91,
58, 16, -7, 48, 55, 40, 9}, {-58, -36, -84, -46, 20, 32, 31,
11, -56, 38, -53, 91, 49, -86}, {-81, -29, 33, 37, -41, -58,
32, -35, -2, 84, 98, -97, 81, 39}, {-35, -64, -97, 13, -26, -79, 49,
24, -49, -48, -62, 99, -13,
20}, {-36, -44, -91, -66, -8, -70, -64, -8, -62, -70, 38, 1, 19,
22}}
smallMatrix = {{992, 461, -757, 484, -177, 976, 716, 277, 311, 825, -899, -848,
221, -205}, {308, 232, -30, 0, -20, 630, 205, 767, 426, 177, 422,
383, -634, -762}, {411, 174, 618, -619, 18, 849, 377, -274,
61, -253, -807, -482, -172, -96}, {-693, 356, -365, 375, 586, 298,
245, 924, -161, -67, 362, 181, 692, -151}, {306, 995, -206, 878,
215, 692, 356, 48, 869, 294, 984, -915, -877, 909}, {-758,
939, -481, -855, -110, 5, -360, -603, -680, 491, 707,
518, -78, -49}, {725, -306, -975, -832, -254, -517,
806, -664, -523, -167, 787, 829, 925, -642}, {-471, 59,
259, -953, -655, 374, 633, 266, -524, -580, -21, 309,
340, -168}, {-605, -464, 305, 252, 364, 37, 72, -761, 57, 853,
600, -109, -718, 884}, {-151, -470, -329, 368, 237,
591, -711, -378, -246, -880, 944, -209, -614,
456}, {-148, -366, -764, 374, 609, -702, 58, -214,
928, -389, -962, -484, 642, -530}, {-249, -336, -395, 74,
166, -820, -486, -870, 637, -420, -593, -862, 344, 273}, {-369,
489, -185, 684, -791, -190, -748, -616, -404, -693,
514, -422, -198, -831}, {-304, 28, 434, -288, 502, 133,
172, -31, -753, 208, 847, -542, -777, 763}}
bigMatrix = basicMatrix.powersMatrix + smallMatrix;
Calculating Det[bigMatrix], and displaying to 10 places, I obtain either 2.284217140 x 10^9769 or 1.142108570 x 10^9769. Neither of these are correct. The reciprocal of the determinant of the inverse gives:
1/Det[Inverse[bigMatrix]] // N[#, 10] &
2.672705588*10^9763
This is the same answer one gets by truncating the elements of bigMatrix and taking the determinant.
bigMatrix2 = N[bigMatrix, 10];
Det[bigMatrix2]
2.672705588*10^9763
Assuming that the answer is not wildly dependent on the precision we might have some reason to trust:
1/Det[Inverse[bigMatrix]]
giving:
2672705587606162693200253937085000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000172614614814380187\
3744406782124594156243273664096717669821188008611374471095367329358081\
1535170005519801715998205770608306130331300000000000000000000000326665\
0674246209123094621358364193355738754341202642780914226469897184310042\
0157114851029953246680074718964934943657745920830494832826636087435261\
7232244329970160000000000000255831989239635286692736634985010028677402\
4012807389350606974012543896094548898376815835808147564254591131229479\
8035312000545131772101187120155441325837715621038171133224482158219460\
2532982693404202950234304828591273519394623033546142866398435725600267\
1738567709607306994543770164338888692431690247858526409746755585856658\
0432245043026227702023943994645915902296798180605554418936597776950816\
9551548598163879398297557606709060904450720588126984121362618171487024\
4004961533285763394707226529491064316333114682961215626234595772846118\
5230669164750702872970802215552641237188496973525030404603055314650955\
4203636533693619143649989208865386899609768556098536757213620019264866\
0684229485175512203421947669909539500368930846401329814419566400401697\
9220169564754600909622330816974811125183198201489490613732945651947188\
5162526402192670889903117187567766999561397064059803380096386790552396\
1336979740604158754859679946166257988690095778204320090101452297792074\
2074618640547070743308188662967776509955727602059893926407614125482022\
0607185409771171015502105996039499131956783496507487988436030908463457\
6088847115120167001705469161062741273884219041284849430551382696380676\
3443652649582683403400225820754304187890284662644860291474064673307934\
3604076510371921112264268843507352921350855058813567741169933593817788\
6181718284991671380078087199867317453420301369645955679171439814313618\
1818754790073027475988223913684279640395370076271071624058688462585446\
1211971182299004751719281949202025134986960478200135644069374120186516\
7031737564051602315254180084128322667926517488354122644960190604633534\
6250160508207334003957275122659277171593771102157191066387536780233220\
8476669612119201659227069595893698322944774535458126162550889842060339\
5733342118949686794845152466056764962162246054692145984677301593012803\
2778862411437059260300890231144715006195015345233009856935102372143717\
9524864947297578990252161606866657671756125476894036757181051200170567\
1501785249452755025006197878951293018537681829796820708977858202296266\
3843274538416839339553185352449689491762727087921772802241821945291996\
9083548224538139751922591252681698727315106604681999203307399118863261\
8767106409959552688190093607634035694746995512006068519689664356309112\
0833231723635686900157516465791063585797849561100560552265615612126509\
6213498662337815787107244897150270670802744446075683796760700950261955\
1962992893836392558095987187341862974361180863954891845189055463764709\
6120628168770332140600534315145068608706814441578895573361630933709645\
2681633988975768756300746485012230561671455031277701666998627921419719\
0316675546119223094497606948999260022493808983282530544574704440665258\
1465008772275420958897250323517324535974903743602262438617749324393901\
1058300002697305355380042461065358965266199531399925371609632492746734\
1511709977639616338814341965171252314759941696766437869267669993256663\
4020506492153455020835521167391940328480334362653676375482289375396049\
0445340741596222516106683680017915504849455567198891167664600196860012\
7353629891225385685034910535448704617845096685396163237932274738020434\
0195847734439239515723172915345707668662624599194154163605444360514034\
4673086296194734513442954830475081165679601422022345110203709820717535\
8852727920415064704619036588359170845775393985192636388982947231911109\
0846938071452757856057512438638290066748655001326861429690651423768338\
5392767500824092819351869960927692073620068607266722381452403994111476\
6723211549043736767741010622607428232834743416370707799251634214522544\
9800721131394512261310698063185838779315500282420222615271434841958001\
4863379099241113772563178729442020837653007589723319093766695830678516\
2219143745925955437338328445739794665365930627263389385179811621133636\
0154922306676636713518505679161390634344890729798727727303390207689122\
1940088411039990838737115970607070162242324251075361488049577438814799\
3315271802038057050087322033061406703975521688477447107623353279153018\
7402139670193198424177958236538734705184050322701648544084556083208587\
1135552623173312351338787872463271244014121083139131568363844148470749\
0049814868213885826295276372300941614401364645848357779953342939096504\
9545395061452657530247825867030521813698179995901662723467807665654813\
3293304929787262393356754514508309593377985270485967382488497059639884\
9198813805418835568287187391024611889351714996208769906627473875029633\
9939781640115811752713710767511480021573812861017649734292097678983443\
9142466012503705553812891534338366699205946210662681457323746249139507\
6607645435061741401733424103634293330175523603725248914259865301413492\
5333147122657732259989368088820481367490144968441881663878197929304616\
7849923201367382228406978313948047597240282356692341479740986581388960\
8540597948196881933737216316672395830514429420309798691738344289718648\
1890228267422523008278311752408175119360170660241954666000996812540544\
7925712110137635012044884912114570125788741250855037878443614319806118\
4786225437297942884167570753082793513889947019334697761455964751633865\
8487108179885522114973300278123330094032232901495345002771109430828373\
0485679061283994395831996917502533327456374363532731172797213263756048\
7346045947621593241719608586645254621551618108994316574792837346805266\
9814220720158895334067798862324807846405704177280316332678905114195363\
6475940728631018390653523864664356274218168664323882644198072000120739\
8721627247534608717384464213095884775945145945893429391987573078754525\
3678014956971556694431190467686223906180991699889437808782448684211457\
0566230695006468388529267576421664189411597190020322627049655527784497\
9072972347949844640981980659424987304899094897884675983729835513536764\
8150813352593435374105201743766477829150923197057688274611573355044386\
0006618872810822831990294942300008823229887480285520652864365013199352\
8070298262656775906327003857880790426131268718509036999336795315748353\
1806038151413444997834686438815749308259886776471671843530003770851147\
8331253057317994586453130505209918391858007488077273584860303197084744\
0519988619719187192435271041678820892412208881211197496271198011935073\
6188830692807734717513447248518996762882053153260227968261907405856561\
5471694962506203081990263094913152759632941388692759852900787635370127\
7891982032051094173630505840693808993568444543076165830374999470482531\
7309760793671580291861056981714983021172104531365929554740359095012607\
9498325261774849018663026549702163138013632259616802317221805613917308\
2380754659845301327274341028656346740589794319090278889796822840434471\
2342744112363268626380136195740740982088243808176812547338081475017899\
0245120819841928304842542061888285597877980345637341953841383063003010\
1178017883159674549295718751961918793943337517968272093722557316678743\
3525173765477589224675162065850467393351083261833876055475431697596122\
6706396487100094570822822107913156638353560488026235276776977362696930\
7318802649809172281522495523148115774674879303213326017492189665459404\
8961655659717438281210121313258117942336179570437892106452997159879264\
6416192603820699115612860027729109082633578603231401410076027883526502\
3898247021664710701493742804630209636015042604178423059116939342584675\
8638516239771162367453552437465310280615694051597062644694956461867446\
7566870081159509132118391213433423380329626605559431904645447505595580\
9450828777173498303817891489386396911977285306347217760300677428028121\
6800288747713355099213078301724777441158434700148416245586870765044383\
6042559948372987791320505500135228978133633692716700370370993506011707\
7575267068683185920738822170750693103731533714364825908792179005142237\
9254362547761382323861590153552296259781620063302526853695682263852507\
1980848788975691428537106576527571991935673077120850576794415438294788\
8828599828021630832834312709140874309139807418823449841598225537904383\
9240309416001235618815134507771400358347730821618745285827289810602391\
4057918646449862311452982223044639631459217745491847379623854095180614\
3823438849583684728187953330697477122137719822986823256260492223460463\
0865582503712647269628284700241907996336618115036794413578008201220432\
7633096451092624022640773355732391421950120131543270135936343919190042\
5939184187663355767187167069119747159059087284982447117781541906458384\
0732525335443413266128667185729842605428024867301876046739944123383676\
2681625581509663603660945080287755799097016649806567545235631750169310\
1605142279133553460727993955277360423698964648116352971060219909220442\
0768617130228037431057066961958520613160306312957533279966102491440300\
4499770827279450047918443367025900978186249906481749466324319072201633\
4157458638447391260979362922627225530149995822506217350199716348992782\
0976981436143788112156945391758333457816340397443665564361600469566510\
5995209075571885257242277817826420496692376865950172240041782130186718\
9634972381589203908584720678978345198588854535399376959832216527085822\
5512593220180437286044149943474143914597182481182493842057351626651723\
5179091500212066112501938424694060652030106711883355699971657979209289\
8896996598004068714977127847253862409102497425915230148643090214488021\
9404986859193306984754661025255771669037402397500434894722185415621673\
3476537432365047803911300619996101726470000000000007171831179178198725\
6753438477942876485294307903666216787369894137378969332590982598060377\
8060197646633025273453610084114853134319482029158379999999999999999999\
9999999999999999999999999999999999999999999999999999999999988579699668\
7284037844559090524468965759586782
Why that would work is a bit of a mystery and maybe it wouldn't always work.
I need to add that this was in Mathematica 9.0.1.