Just a quick update on the problems the authors encountered in the Durán et al. article:

As we all know, there's an infinite amount of math computations out there, and we do a tremendous amount of testing but it's always possible to miss something. We thank the article's authors for pointing out these particular bugs which have now been fixed with last week's Mathematica 10.0.2 release.

I should note that we ran a blog post in 2007 by Stephen Wolfram that addresses these kinds of issues, and is worth revisiting:

Mathematics, Mathematica, and Certainty

A relevant extract:

"We run millions and millions of tests on every version of Mathematica, trying to exercise every part of the system. And doing that is orders of magnitude more powerful at catching bugs than any kind of pure human testing.

Sometimes we use the symbolic capabilities of Mathematica to analyse the raw code of Mathematica. But pretty quickly we tend to run right up against undecidability: there’s a theoretical limit to what we can automatically verify."

If folks here are interested in more details surrounding the particular issues in question, I could try and see about getting one of our senior developers to write something up (no promises, as we are all pretty busy).

My Windows 8.1 64 bit computer running Mathematica 10.0.0.0 gives the same value 26727055876061626.... for Det[bigMatrix] obtained previously by Udo Krause and David Park Jr. I have also written a program that diagonalizes an integer matrix by row and column reductions. If the matrix is square the product of the diagonal entries is then the determinant. This gives the same answer, so there is little doubt we have the correct value. Mathematica 9.0.0.0 running on the same computer alternates at random between two values for Det[bigMatrix], both incorrect, and one exactly twice the other. These appear to be the values 2.284217140 x 10^9769 and 1.142108570 x 10^9769 mentioned by David Park Jr.

I agree with you Ilian that it is (nearly) impossible to make bug-free software, there will always be something that goes wrong. But the more important point is, is that they reported this to Wolfram before the release of v10. Unfortunately it was not fixed...

It would also be nice if one could check the status of a certain Case online and see if it is resolved or not, like a bugtracker. I can imagine Wolfram is already using something like that; it would be nice to see this available also to the outside. So one knows if a bug is already known or not, one could then 'upvote' those bugs rather than resubmitting them...

With Mathematica 10.0.1 (Windows 7 64 Bit Home Premium) this works out of the box

In[4]:= bigMatrix = basicMatrix.powersMatrix + smallMatrix;

In[7]:= Det[bigMatrix]
Out[7]= 26727055876061626932002539370850000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000000000001726146148\
1438018737444067821245941562432736640967176698211880086113744710953673\
2935808115351700055198017159982057706083061303313000000000000000000000\
0032666506742462091230946213583641933557387543412026427809142264698971\
8431004201571148510299532466800747189649349436577459208304948328266360\
8743526172322443299701600000000000002558319892396352866927366349850100\
2867740240128073893506069740125438960945488983768158358081475642545911\
3122947980353120005451317721011871201554413258377156210381711332244821\
5821946025329826934042029502343048285912735193946230335461428663984357\
2560026717385677096073069945437701643388886924316902478585264097467555\
8585665804322450430262277020239439946459159022967981806055544189365977\
7695081695515485981638793982975576067090609044507205881269841213626181\
7148702440049615332857633947072265294910643163331146829612156262345957\
7284611852306691647507028729708022155526412371884969735250304046030553\
1465095542036365336936191436499892088653868996097685560985367572136200\
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0040169792201695647546009096223308169748111251831982014894906137329456\
5194718851625264021926708899031171875677669995613970640598033800963867\
9055239613369797406041587548596799461662579886900957782043200901014522\
9779207420746186405470707433081886629677765099557276020598939264076141\
2548202206071854097711710155021059960394991319567834965074879884360309\
0846345760888471151201670017054691610627412738842190412848494305513826\
9638067634436526495826834034002258207543041878902846626448602914740646\
7330793436040765103719211122642688435073529213508550588135677411699335\
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6258544612119711822990047517192819492020251349869604782001356440693741\
2018651670317375640516023152541800841283226679265174883541226449601906\
0463353462501605082073340039572751226592771715937711021571910663875367\
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5630911208332317236356869001575164657910635857978495611005605522656156\
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3370964526816339889757687563007464850122305616714550312777016669986279\
2141971903166755461192230944976069489992600224938089832825305445747044\
4066525814650087722754209588972503235173245359749037436022624386177493\
2439390110583000026973053553800424610653589652661995313999253716096324\
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3802043401958477344392395157231729153457076686626245991941541636054443\
6051403446730862961947345134429548304750811656796014220223451102037098\
2071753588527279204150647046190365883591708457753939851926363889829472\
3191110908469380714527578560575124386382900667486550013268614296906514\
2376833853927675008240928193518699609276920736200686072667223814524039\
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2113363601549223066766367135185056791613906343448907297987277273033902\
0768912219400884110399908387371159706070701622423242510753614880495774\
3881479933152718020380570500873220330614067039755216884774471076233532\
7915301874021396701931984241779582365387347051840503227016485440845560\
8320858711355526231733123513387878724632712440141210831391315683638441\
4847074900498148682138858262952763723009416144013646458483577799533429\
3909650495453950614526575302478258670305218136981799959016627234678076\
6565481332933049297872623933567545145083095933779852704859673824884970\
5963988491988138054188355682871873910246118893517149962087699066274738\
7502963399397816401158117527137107675114800215738128610176497342920976\
7898344391424660125037055538128915343383666992059462106626814573237462\
4913950766076454350617414017334241036342933301755236037252489142598653\
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2930461678499232013673822284069783139480475972402823566923414797409865\
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8971864818902282674225230082783117524081751193601706602419546660009968\
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1980611847862254372979428841675707530827935138899470193346977614559647\
5163386584871081798855221149733002781233300940322329014953450027711094\
3082837304856790612839943958319969175025333274563743635327311727972132\
6375604873460459476215932417196085866452546215516181089943165747928373\
4680526698142207201588953340677988623248078464057041772803163326789051\
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0012073987216272475346087173844642130958847759451459458934293919875730\
7875452536780149569715566944311904676862239061809916998894378087824486\
8421145705662306950064683885292675764216641894115971900203226270496555\
2778449790729723479498446409819806594249873048990948978846759837298355\
1353676481508133525934353741052017437664778291509231970576882746115733\
5504438600066188728108228319902949423000088232298874802855206528643650\
1319935280702982626567759063270038578807904261312687185090369993367953\
1574835318060381514134449978346864388157493082598867764716718435300037\
7085114783312530573179945864531305052099183918580074880772735848603031\
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3537012778919820320510941736305058406938089935684445430761658303749994\
7048253173097607936715802918610569817149830211721045313659295547403590\
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7501789902451208198419283048425420618882855978779803456373419538413830\
6300301011780178831596745492957187519619187939433375179682720937225573\
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2802812168002887477133550992130783017247774411584347001484162455868707\
6504438360425599483729877913205055001352289781336336927167003703709935\
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0514223792543625477613823238615901535522962597816200633025268536956822\
6385250719808487889756914285371065765275719919356730771208505767944154\
3829478888285998280216308328343127091408743091398074188234498415982255\
3790438392403094160012356188151345077714003583477308216187452858272898\
1060239140579186464498623114529822230446396314592177454918473796238540\
9518061438234388495836847281879533306974771221377198229868232562604922\
2346046308655825037126472696282847002419079963366181150367944135780082\
0122043276330964510926240226407733557323914219501201315432701359363439\
1919004259391841876633557671871670691197471590590872849824471177815419\
0645838407325253354434132661286671857298426054280248673018760467399441\
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5016931016051422791335534607279939552773604236989646481163529710602199\
0922044207686171302280374310570669619585206131603063129575332799661024\
9144030044997708272794500479184433670259009781862499064817494663243190\
7220163341574586384473912609793629226272255301499958225062173501997163\
4899278209769814361437881121569453917583334578163403974436655643616004\
6956651059952090755718852572422778178264204966923768659501722400417821\
3018671896349723815892039085847206789783451985888545353993769598322165\
2708582255125932201804372860441499434741439145971824811824938420573516\
2665172351790915002120661125019384246940606520301067118833556999716579\
7920928988969965980040687149771278472538624091024974259152301486430902\
1448802194049868591933069847546610252557716690374023975004348947221854\
1562167334765374323650478039113006199961017264700000000000071718311791\
7819872567534384779428764852943079036662167873698941373789693325909825\
9806037780601976466330252734536100841148531343194820291583799999999999\
9999999999999999999999999999999999999999999999999999999999999999999885\
796996687284037844559090524468965759586782

In[10]:= 1/Det[Inverse[bigMatrix]] == Det[bigMatrix]
Out[10]= True

the determinant given by D J M Park Jr agrees with the one above.