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Determinants of the classical test matrices
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Numbers
$A_n$
$n$ 
$\det(A_n)$
Hilbert matrix
2:
1/12
Hilbert matrix
3:
1/2160
Hilbert matrix
4:
1/6048000
Hilbert matrix
5:
1/266716800000
Hilbert matrix
6:
1/186313420339200000
Hilbert matrix
7:
1/2067909047925770649600000
Hilbert matrix
8:
1/365356847125734485878112256000000
Hilbert matrix
9:
1/1028781784378569697887052962909388800000000
Hilbert matrix
10:
1/46206893947914691316295628839036278726983680000000000
Hilbert matrix
11:
1/33122504897063413755362143627040727106080127672469422080000000000
Hilbert matrix
12:
1/379106579436304517151885479034796391880188687864118464104324304732160000000000
Hilbert matrix
13:
1/69305039341130527126879829549184590532766990585717637092894872077560293196038144000000000000
Hilbert matrix
14:
1/202416246188391670717379100929711579882350239090925841863293007224574286736535499564572999680000000000000000
Hilbert matrix
15:
1/9446949653634668571373109351236989087975627994978804269595338137635022705891424600259116300098090513203200000000000000000000
Hilbert matrix
16:
1/7046525047976572667724995315589896337917049049168943848740328272995413909271403490510630224928824925252695730373189959680000000000000000000000
Hilbert matrix
17:
1/84014463604874440715352002945844264431345730525198324022328094341880570822819028414402392963500535432692716170056914136738952052236877824000000000000000000000000
Hilbert matrix
18:
1/16013167655630990153779179670990515462750657017396108207154121114316054829393635304279105679663156225153608544807387972095707264535982067156578872858771456000000000000000000000000000
Hilbert matrix
19:
1/48796108907551903221120343890645829271034905772887829316663417404408237089424172111058257913991344176537643387280192336309618529400584258799545334919201208467089170869780480000000000000000000000000000000
Hilbert matrix
20:
1/2377454716768534509091644243427616440175419837753486493033185331234419759310644585187585766816573773440565759867265558971765638419710793303386582324149811241023554489166154717809635257797836800000000000000000000000000000000000
Hilbert matrix
21:
1/1852199570799563504779826534318615410359478694679745750316417111878996533152617768892344310944722927682345063938230785024474479390959212535885051685075157842022058776465680568137293461758448523921333107377171333120000000000000000000000000000000000000
Hilbert matrix
22:
1/23074750769641488684469152844309942442271672562969235831981894249241371297478253331906249285131254014680117650878698037168694450592177334203551864489666331676901432792363674686536272499287124436530632096418053762490392522390626501656576000000000000000000000000000000000000000
Hilbert matrix
23:
1/4597077841329975772431123126143764281158459635117402179429258553523719563115752031800142342257426550819650876769161532475466271612113635848897623107578859270294977173209948842945189375715087257244809452599995332188610751112991186013377015157344261247837667328000000000000000000000000000000000000000000
Hilbert matrix
24:
1/14646753997821840157193087382630089897761438432760168157772093615197744878068295275034427683794161439808660205654182552055373504374290585122908288608443758340856946542009952245901666087877873465247614840766479180831884382399001259187948618047888133828032527999451190315416289447772160000000000000000000000000000000000000000000000
Hilbert matrix
25:
1/746332515470541859307349018107431789872329747913727730760621191984777569063589205517303938509441565299809996599554994981363939681729476785432795097905555120762172958990784084457536372476333251919350316666089029356479086324949498677522696089012012028938712165715166669721997554754957779660912439584392373862400000000000000000000000000000000000000000000000000
Hilbert matrix
26:
1/608232424841085072418678538082910899807644656052162410814141027095169069739936305679443103880482874766617530535227681963416841993625722129726973691819377418476517657993936301945313004357624240738883495522538123643054851209361655702797275468112112346743285163773324116631700045450458392960377126661147866856068257133509227218556954450329600000000000000000000000000000000000000000000000000
Hilbert matrix
27:
1/7928045161872069809758376074941151073507548198441160117266368751763856532942313551030442061303837648382489381822982584515911354313284651516949235032066865913308841801265920069503296143729147995128385530303432705225312524018836038118449514009827090882199156550245512783776473037857212786046102043488905464528515214333126681887839737720255758222291998459238604478401740800000000000000000000000000000000000000000000000000
Hilbert matrix
28:
1/1652850921775258983316633014900525443396107807534646089721970045557198144474939450686557584194393271314097263480907699234458237043493947706780732049042012250307330605560385652010431618112617005285921021075510790458213441811135497211414243201544927838856086981876472049610477909034226777109409988260876652790294545226074484215406158215028691583863226516659709649759574521717681018692541578361738100736000000000000000000000000000000000000000000000000000
Hilbert matrix
29:
1/5511663906275075184277453668311482567906819669849714800353229530341415728961797449357542773962044920835447904660821060181620705158183947450978250430725869247760987066376912226763060763227409706408096849386775886378369891692420207048670607628175843808684027663336844861149117264743847816060486257511116237172867779517235063217873340507278072486570782239069342935915319126702457057009830451722030158137092794887587931390093837939507200000000000000000000000000000000000000000000000000000
Hilbert matrix
30:
1/293983281023955245055445414967145766270470909701745379023785086313360837488683725402224423670871543240867850801733611107677638030185935357884574159309950862445430661887957338647852712385421925367043913854648814370216926583441155883157602140414022772120159075252070534506237128214455820882780530921346148916982305134960099240772086154178279059855503658572588997750723108283007463683509421712230491766353997894562500785567662805738963174251576696435327803921719623680000000000000000000000000000000000000000000000000000000
Lehmer matrix
2:
3/4
Lehmer matrix
3:
5/12
Lehmer matrix
4:
35/192
Lehmer matrix
5:
21/320
Lehmer matrix
6:
77/3840
Lehmer matrix
7:
143/26880
Lehmer matrix
8:
143/114688
Lehmer matrix
9:
2431/9289728
Lehmer matrix
10:
46189/928972800
Lehmer matrix
11:
4199/486604800
Lehmer matrix
12:
96577/70071091200
Lehmer matrix
13:
7429/36436967424
Lehmer matrix
14:
7429/264505393152
Lehmer matrix
15:
215441/59513713459200
Lehmer matrix
16:
6678671/15235510645555200
Lehmer matrix
17:
392863/7848596393164800
Lehmer matrix
18:
392863/72655578039582720
Lehmer matrix
19:
765049/1380455982752071680
Lehmer matrix
20:
765049/14158522900021248000
Lehmer matrix
21:
31367009/6243908598909370368000
Lehmer matrix
22:
1348781387/3022051761872135258112000
Lehmer matrix
23:
58642669/1544604233845758020812800
Lehmer matrix
24:
2756205443/889692038695156619988172800
Lehmer matrix
25:
2756205443/11348112738458630356992000000
Lehmer matrix
26:
2756205443/150418121788196747476992000000
Lehmer matrix
27:
146078888479/109654810783595428910727168000000
Lehmer matrix
28:
146078888479/1563079484624342113927456358400000
Lehmer matrix
29:
5037203051/795250965861507391296425164800000
Lehmer matrix
30:
297194980009/715725869275356652166782648320000000
Vandermonde matrix
2:
1
Vandermonde matrix
3:
2
Vandermonde matrix
4:
12
Vandermonde matrix
5:
288
Vandermonde matrix
6:
34560
Vandermonde matrix
7:
24883200
Vandermonde matrix
8:
125411328000
Vandermonde matrix
9:
5056584744960000
Vandermonde matrix
10:
1834933472251084800000
Vandermonde matrix
11:
6658606584104736522240000000
Vandermonde matrix
12:
265790267296391946810949632000000000
Vandermonde matrix
13:
127313963299399416749559771247411200000000000
Vandermonde matrix
14:
792786697595796795607377086400871488552960000000000000
Vandermonde matrix
15:
69113789582492712943486800506462734562847413501952000000000000000
Vandermonde matrix
16:
90378331112371142262979521568630736335023247731599748366336000000000000000000
Vandermonde matrix
17:
1890966832292234727042877370627225068196418587883634153182519380410368000000000000000000000
Vandermonde matrix
18:
672593129192865130334217631473916658864122332882577979675277211683839238972899328000000000000000000000000
Vandermonde matrix
19:
4306192564997715382115598640379294845786123319603755168023536027873932927153136831171640950784000000000000000000000000000
Vandermonde matrix
20:
523827226948912906162136183269887782788685420217963126789982275317725639664591791615428617583779071590924288000000000000000000000000000000
Vandermonde matrix
21:
1274420312381610764187232669591245728094053576480399681464607478621086015845192873311847341571189522511934190056991170232320000000000000000000000000000000000
Vandermonde matrix
22:
65111334482340755583337032079153276047018894694274871209973336023462275586396482041798135409436085152342597056410325664391109570976337100800000000000000000000000000000000000000
Vandermonde matrix
23:
73185187344722251685917129061596359994436879170423331597220791224873045503248922811481956548134913475324429128068839519292427638045690327208977109451014144000000000000000000000000000000000000000000
Vandermonde matrix
24:
1891984688274192607739481159411532607771433686902374691017403396401535783431356418138933900060697141172863325793213721068660204501893619736331984249640376781921613013749596160000000000000000000000000000000000000000000000
Vandermonde matrix
25:
1173878875943484045145854663582065069715341161356797835752331993963556125154689322181575091217882971654578156844286587738831291224576726252820647026675651753423450654917092461536374019784808857600000000000000000000000000000000000000000000000000
Vandermonde matrix
26:
18208281810188658276327614252643415193394793745060606890429618992193283129810291626991262221974029109363817787141461966252773116546051694612770469960217933012913302274713118105435774192291006757570987304784651878400000000000000000000000000000000000000000000000000000000
Vandermonde matrix
27:
7343244575835979773361426394615625529240080651911303827118013756998588507133433307910669924513634706799292902446390806879822780229775160924771413690082976195597708144958301470585112704676289169327783424988568398136986317713011480985600000000000000000000000000000000000000000000000000000000000000
Vandermonde matrix
28:
79959631528770670600322037454897824976792430707106611658531105568420623057823058204406300276129173697769238298117257446827525920399207590372612824866949329475045338094293229818656245740234951168513312204538663055609170465072773314478245835098717506292940800000000000000000000000000000000000000000000000000000000000000000000
Vandermonde matrix
29:
24378759692569493001843754884245889709720110123963670883808668267742801150073945462248212671568308430773839473807128215512114800359197934772179842358860234182649345336155492659539815888875032454531949849910359883836215115120785688966546366708482811138722465511411531795422982963200000000000000000000000000000000000000000000000000000000000000000000000000
Vandermonde matrix
30:
215551190904274323928924367943222430700137095451739175565450519136588260899040178316841478424566695269089767808576345615153945752404719288134605257465113604136602603261108922442719744278826536334531599709796863580066482222372814044233417149800595451399256699593376354273158453301367277680488410399322931200000000000000000000000000000000000000000000000000000000000000000000000000000000
Cauchy matrix
2:
1/72
Cauchy matrix
3:
1/43200
Cauchy matrix
4:
1/423360000
Cauchy matrix
5:
1/67212633600000
Cauchy matrix
6:
1/172153600393420800000
Cauchy matrix
7:
1/7097063852481244869427200000
Cauchy matrix
8:
1/4702142622508202833251304734720000000
Cauchy matrix
9:
1/50019370356486058711268515056654483456000000000
Cauchy matrix
10:
1/8537000898240926708833515201784986712482596782080000000000
Cauchy matrix
11:
1/23365674874545238092272627703110594203896316624245449356738560000000000
Cauchy matrix
12:
1/1025163331422159477883374029444818871881163521419883129418493194587298856960000000000
Cauchy matrix
13:
1/720813992171362160435826355209249252295096362285814856348362406529873585414714320486400000000000000
Cauchy matrix
14:
1/8120251581841233297500810440356867565508291601515035627692980253625356831294900021832149198962688000000000000000000
Cauchy matrix
15:
1/1465387401836668774813339717252690679593841235023684570865040255471363387281567173259185477862791557143607640064000000000000000000000
Cauchy matrix
16:
1/4235528023982527009979480597042947970854751630123597944488937527060109860796279825923490914746005808312611198164051866508538675200000000000000000000000
Cauchy matrix
17:
1/196056674838298617172366683563880498628313159724046675672080259836959306569281002642206161802509082459062113803139392583499949025381743003466465280000000000000000000000000
Cauchy matrix
18:
1/145321663056435042518513861837248412841204322596764135673364077065004964538965657345189553005941601168306260827363158286361167974856928977819639556314848754977996800000000000000000000000000000
Cauchy matrix
19:
1/1724711341750951831542568286361595187954490443550863215006842223968320422818641553361957023138461770836316776153743962641601785399031706501397401182778518398548060362515666513690624000000000000000000000000000000000
Cauchy matrix
20:
1/327723880133278729476665890222832025204546816520485699917379710978406689527792232144837486511702636314231098574648221249394901642708363125160341523947219537233313370189682724077124853837717640184856576000000000000000000000000000000000000
Cauchy matrix
21:
1/996961004017253343362586210554249644503921358504785684084937632076432348414857032405768401997732991389032889630017997801446216655003291492301690144178094731694290668613164318213591168820611927198052252148036525144872373452800000000000000000000000000000000000000
Cauchy matrix
22:
1/48551559182499928776909650558018840391276672162454725560313395723631396743175744684899576333036662376472525216501549287429448036316315069236165589615980795129942114598995633837312166801941572965642791906109798045555281912818154119177170424923422720000000000000000000000000000000000000000
Cauchy matrix
23:
1/37849721955975299775697954101371039727621387284459903137600066005067321906402480374205424302796131712818596494695495573571409293828583997781270600680543580356202778973847380329528988688613660646714619862643737279658277116969417530470505661240049189369468569235061787852800000000000000000000000000000000000000000000
Cauchy matrix
24:
1/472322718165619222030719287637962225951935864449430690383546707955848791864809319124538537655221202628649749807130220710580620094806527033759151393913168715231824814518339944921793684215876880620914744192591334058813745731763401819550829570494632516609647451941706079594097386305713171905642496000000000000000000000000000000000000000000000000
Cauchy matrix
25:
1/94344345884844122895669369627976073035990346945793649973185107411161523314255514417225236246776078681522617448130111877369572484796897612065933522376847473280302030846720901675566068651230469157335570350251653672118116107520612044141135536448753640214618431419605770076705202431236268020442693962925170374671475057413324800000000000000000000000000000000000000000000000000
Cauchy matrix
26:
1/301633731818658837322516871486422064389966375958260427178209133340538833710273010182765257813835295266902143846224433495632064165798183037538636008132348723633543321391691253376911353021101657158322671560064076829998738985699933604221276362190679626834971123672542113594333159479105944855750763019479906862133762527154162831989840372221044890322495078400000000000000000000000000000000000000000000000000
Cauchy matrix
27:
1/15435423693967500724922063012821802214575177927185942253031613715318831358472378088491225744516150286670908883083515123114479244333454028952426063471475123097912536904779054769716765417762665023679747290965933549453582838407519075790508041034846752095548624115263842018168635526172662587278959829231525886598186805923994473078740013500917144859598605635297006246395960577856949033369600000000000000000000000000000000000000000000000000
Cauchy matrix
28:
1/12642145309840652653214265705992609303815597432252992941944835173403088111426427069019959496466450236328169805910234121643649364572995370759099879078770717983384034509465735954599399084615576049408132901742287054442968270670522882538130212181218442138919692176840664911516540449743537249491026759142157333429690701149802366289941538298166191000830190571678816569174059510324693510767060414379145928610696402923683840000000000000000000000000000000000000000000000000000
Cauchy matrix
29:
1/165720667525874074606782800879947971243750456141446875895540621319675064484740636065889185812508927688193194659436861754970197710128084765191158636882273521298898181477855426745209032688056753379368153769469492021517612708279917749826404452046238142581050088772487572372960926721331327661106784030515237842806520187795088592828915087663099846700821036547128119817615055981242347671528905620856432901622769637736343766148059528344176393208083775488000000000000000000000000000000000000000000000000000000
Cauchy matrix
30:
1/34767809717363132770618111603748478866514045559675119423637442347693427872336185915903600949727565688512425687551728790953230983654612947880686640340848775139119194123686029568357518339851752172038585284270284784110742006570826177503344906467148331128961947316254145499233896924560136924601181973306894033338133775362887249764402725843635016814711320344415326722721639484313759086848749307244658949958839254038310872532906539606871535978838817484593983322127315324008209240628920320000000000000000000000000000000000000000000000000000000
Fiedler matrix
2:
-1
Fiedler matrix
3:
4
Fiedler matrix
4:
-12
Fiedler matrix
5:
32
Fiedler matrix
6:
-80
Fiedler matrix
7:
192
Fiedler matrix
8:
-448
Fiedler matrix
9:
1024
Fiedler matrix
10:
-2304
Fiedler matrix
11:
5120
Fiedler matrix
12:
-11264
Fiedler matrix
13:
24576
Fiedler matrix
14:
-53248
Fiedler matrix
15:
114688
Fiedler matrix
16:
-245760
Fiedler matrix
17:
524288
Fiedler matrix
18:
-1114112
Fiedler matrix
19:
2359296
Fiedler matrix
20:
-4980736
Fiedler matrix
21:
10485760
Fiedler matrix
22:
-22020096
Fiedler matrix
23:
46137344
Fiedler matrix
24:
-96468992
Fiedler matrix
25:
201326592
Fiedler matrix
26:
-419430400
Fiedler matrix
27:
872415232
Fiedler matrix
28:
-1811939328
Fiedler matrix
29:
3758096384
Fiedler matrix
30:
-7784628224
Kac-Murdock-Szegő matrix
2:
3/4
Kac-Murdock-Szegő matrix
3:
9/16
Kac-Murdock-Szegő matrix
4:
27/64
Kac-Murdock-Szegő matrix
5:
81/256
Kac-Murdock-Szegő matrix
6:
243/1024
Kac-Murdock-Szegő matrix
7:
729/4096
Kac-Murdock-Szegő matrix
8:
2187/16384
Kac-Murdock-Szegő matrix
9:
6561/65536
Kac-Murdock-Szegő matrix
10:
19683/262144
Kac-Murdock-Szegő matrix
11:
59049/1048576
Kac-Murdock-Szegő matrix
12:
177147/4194304
Kac-Murdock-Szegő matrix
13:
531441/16777216
Kac-Murdock-Szegő matrix
14:
1594323/67108864
Kac-Murdock-Szegő matrix
15:
4782969/268435456
Kac-Murdock-Szegő matrix
16:
14348907/1073741824
Kac-Murdock-Szegő matrix
17:
43046721/4294967296
Kac-Murdock-Szegő matrix
18:
129140163/17179869184
Kac-Murdock-Szegő matrix
19:
387420489/68719476736
Kac-Murdock-Szegő matrix
20:
1162261467/274877906944
Kac-Murdock-Szegő matrix
21:
3486784401/1099511627776
Kac-Murdock-Szegő matrix
22:
10460353203/4398046511104
Kac-Murdock-Szegő matrix
23:
31381059609/17592186044416
Kac-Murdock-Szegő matrix
24:
94143178827/70368744177664
Kac-Murdock-Szegő matrix
25:
282429536481/281474976710656
Kac-Murdock-Szegő matrix
26:
847288609443/1125899906842624
Kac-Murdock-Szegő matrix
27:
2541865828329/4503599627370496
Kac-Murdock-Szegő matrix
28:
7625597484987/18014398509481984
Kac-Murdock-Szegő matrix
29:
22876792454961/72057594037927936
Kac-Murdock-Szegő matrix
30:
68630377364883/288230376151711744
Grcar matrix
2:
2
Grcar matrix
3:
4
Grcar matrix
4:
8
Grcar matrix
5:
15
Grcar matrix
6:
29
Grcar matrix
7:
56
Grcar matrix
8:
108
Grcar matrix
9:
208
Grcar matrix
10:
401
Grcar matrix
11:
773
Grcar matrix
12:
1490
Grcar matrix
13:
2872
Grcar matrix
14:
5536
Grcar matrix
15:
10671
Grcar matrix
16:
20569
Grcar matrix
17:
39648
Grcar matrix
18:
76424
Grcar matrix
19:
147312
Grcar matrix
20:
283953
Grcar matrix
21:
547337
Grcar matrix
22:
1055026
Grcar matrix
23:
2033628
Grcar matrix
24:
3919944
Grcar matrix
25:
7555935
Grcar matrix
26:
14564533
Grcar matrix
27:
28074040
Grcar matrix
28:
54114452
Grcar matrix
29:
104308960
Grcar matrix
30:
201061985
Wilkinson matrix
2:
-3/4
Wilkinson matrix
3:
-2
Wilkinson matrix
4:
-35/16
Wilkinson matrix
5:
-4
Wilkinson matrix
6:
-403/64
Wilkinson matrix
7:
-20
Wilkinson matrix
8:
-13419/256
Wilkinson matrix
9:
-252
Wilkinson matrix
10:
-925699/1024
Wilkinson matrix
11:
-5610
Wilkinson matrix
12:
-102416243/4096
Wilkinson matrix
13:
-187944
Wilkinson matrix
14:
-16310521875/16384
Wilkinson matrix
15:
-8760272
Wilkinson matrix
16:
-3516410439803/65536
Wilkinson matrix
17:
-540315160
Wilkinson matrix
18:
-983933221071619/262144
Wilkinson matrix
19:
-42535905530
Wilkinson matrix
20:
-346315359485263299/1048576
Wilkinson matrix
21:
-4158250120140
Wilkinson matrix
22:
-149639629352069482003/4194304
Wilkinson matrix
23:
-493938370048692
Wilkinson matrix
24:
-77840328577009611740555/16777216
Wilkinson matrix
25:
-70043599065844404
Wilkinson matrix
26:
-47970016327489759564744323/67108864
Wilkinson matrix
27:
-11684931733886455730
Wilkinson matrix
28:
-34553998333395765978509862419/268435456
Wilkinson matrix
29:
-2264985487707963662992
Wilkinson matrix
30:
-28761989163249230359209430481299/1073741824
Clement matrix
2:
-1
Clement matrix
4:
9
Clement matrix
6:
-225
Clement matrix
8:
11025
Clement matrix
10:
-893025
Clement matrix
12:
108056025
Clement matrix
14:
-18261468225
Clement matrix
16:
4108830350625
Clement matrix
18:
-1187451971330625
Clement matrix
20:
428670161650355625
Clement matrix
22:
-189043541287806830625
Clement matrix
24:
100004033341249813400625
Clement matrix
26:
-62502520838281133375390625
Clement matrix
28:
45564337691106946230659765625
Clement matrix
30:
-38319607998220941779984862890625
Parter matrix
2:
16/3
Parter matrix
3:
2048/135
Parter matrix
4:
1048576/23625
Parter matrix
5:
34359738368/260465625
Parter matrix
6:
4503599627370496/11371668721875
Parter matrix
7:
9444732965739290427392/7888446990683634375
Parter matrix
8:
79228162514264337593543950336/21776965089186101310140625
Parter matrix
9:
170141183460469231731687303715884105728/15330043202319289712414934678515625
Parter matrix
10:
1461501637330902918203684832716283019655932542976/43033523436556282747812223470803609794921875
Parter matrix
11:
200867255532373784442745261542645325315275374222849104412672/1927983533652930855481078826533672813447199742802734375
Parter matrix
12:
110427941548649020598956093796432407239217743554726184882600387580788736/344795425404576705797704088613314881404393134165837604181728515625
Parter matrix
13:
3885337784451458141838923813647037813284813678104279042503624819477808570410416996352/3939535380432660481737902361441262572314314558395701458999546507445773681640625
Parter matrix
14:
546812681195752981093125556779405341338292357723303109106442651602488249799843980805878294255763456/179781741661237211642347145472441720566037161335982334498678098832668480910022434539794921875
Parter matrix
15:
1231312693637327475383720003129487931408741852202045208373384168882678805359287831606695820465153613775207124697088/131102762724248208694626102055982701077513741180474774872190798192181686391991410348943862541866302490234375
Parter matrix
16:
11090678776483259438313656736572334813745748301503266300681918322458485231222502492159897624416558312389564843845614287315896631296/381992888274486172330023966308185542489601516762345685811779018531247003918774605768603887469672985625462095737457275390625
Parter matrix
17:
102293456496754433437912178025862473506770063938845774671352855253004181137646079840102190385184504910965208878986252219038039267058918532916516487168/1138608299507615162647427292330629670698783697607908542468662932849770354828400457047052852766179259956360889577703449255507051944732666015625
Parter matrix
18:
3773962424821541352241554580988268890916921220416440428376206300245624162392148852086126725177658767541468375030763844899770584629924792632561434251432696043649395326976/13563680594226352698868417457008363251850061118693602687589871286839911501626127259714664607510903154430496578059712351350009807499277087129704654216766357421875
Parter matrix
19:
2227754207823337509102134573095845373483021732054960903603121346630505452738612005129840239901060253798165190221481644194672219102234100585084307285020612396607419274589973120157653414182912/2583244496406594136328099897535836001132395518344940923334796681624438664057934161269921018022327444953972978976807115134633782808793490169963532250695127382553182542324066162109375
Parter matrix
20:
5260135901548373507240989882880128665550339802823173859498280903068732154297080822113666536277588451226982968856178217713019432250183803863127814770651880849955223671128444598191663757884322717271293251735781376/1966584448933681291700111990511249252996867781069892652927671711453458253490231480360206319125082095026615600393917412850475986162580856819591467906737908092130755099294776354069472290575504302978515625
Parter matrix
21:
794889263257962974796277498092801308291525640763748664903194643469338087775424965801409745320266996710649718116931109481559848982586784968419475084821084743272680947722675151641735826243378403750534655587182832000457137589153821622272/95756473471527368203359098038636302734320254570510484032372333543233950174955829493086940570565074629250599717033100754347686744918373037370505012065439012316754988621149326723236821614126773416110445396043360233306884765625
Parter matrix
22:
480481077043500814718154092512592439123952613987168226347385561008808420007630829308634252709141208374307457227821149607627692202643343568752733498024953930242542523045817764949544214392905306388478705146745768073877141698859815495632935288783334250628775936/18639635230605950582990575619102336720656612652585253503322680860780678791015192710857351144273949000087985951216766601554775840266484387042002198480751269293590050624585780648624718114741661821832371369282735343395152243829215876758098602294921875
Parter matrix
23:
4646927838993072071243945306718439250410188130185607684549287060362314553626263667328650982800488595593121011844353040782670578892327830336846345565944983205571783876398312106070895030732180427719497486819848241268961714708993375275490934188589556509412640954544199727574266715045888/58023279796528313036778874394104795804150278629297979673062087444214169731311277950493607523991494725719922634357275601790646250234277201850765042424959270476057390617640003320419494644082324993085387906625818657872956700448451188652568700755864483653567731380462646484375
Parter matrix
24:
179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216/722140726079450649978046196471342605407045834027578376367384992090511968103289122311552762289829460554916410826200404806737182623725342065933789973823777935755001596224543168482663038365776607387156022411318741208938681865036766463691667048954725551639913210973715109730619587935507297515869140625
Parter matrix
25:
1780349047995611020364810064179058152051171455050376216246379701505639176934839875954985151130181381953759250081886684246840275834402888741179300032387564007835010793394194393567443569103953032743070880304493124191885516573524617665361384050812922151669040379634673121374356636181347188484885430329302536482647592003015875329283244163072/2299814624023306440059372328309540401870462162098782696355188655078330830543381562183319794476727956275051550071401517193197980470453370256239829330997598599886249228660944979630934777555149960305009693579987739018097578513714869135288072761419420858120304547144634856878739999583757563296959602894276031292974948883056640625
Parter matrix
26:
70526891853356034531603979043031594940743371757357833919419337999977433871326190057052096664518845125780975284354914663582043862183185043558365634079656826243725326219902938648103648856146145787520889095860455531443591926858713371483719982145691511131075312745428721576814913895557944855745498204068445694181321575408727502406563906376986273529501741633684526596096/29285326003717614607804626715592039854049939876465617875516393067336200504839511603258277578530907832960527472351549373766632209825879529344513468164156788136301696184084726853654890040083335749409558712620378277410078995162362596998825850361411534524766905791872988667930912618485280210592786529340065206396141767506957620526009122841060161590576171875
Parter matrix
27:
44701728395069099752361331922969081273299096478473962528531634388733189016699098922387382209961629959968059745377174167657913607628398696609864543063878460605660905572095226320033634320608234016852926043587438147506638476118703283308151078304517051284125857647673490542368762712320361057970140648754594093324662407843349606302070123278473831211371852057741278886313324477610627095975002843906048/5964398272540535029373394364875651916634361710058601930960526124478747871090035862673372190515166935687886994684251423287559381642304349511073776331120074272799787226239922242727982724059384648164400666593400970516286320507430210211584256581236940892518782856849779672531137274539963887482758571714650048101771690544028853874785538460366673272311910380949484533630311489105224609375
Parter matrix
28:
113332345662497260120030995197439525395792673164032384708366964034128343191802752604049236479624647906219612291751933107682859193847720375112403477223230164721340718453679923789759197026541777701436695974840326452569016075159000098460717975752985180146622027746749209964281935326798053887506960149734146070787784228703303325736089316209641122500124748224789789664849837909153579611548122073831823450157992271712195659585028096/4857292303245568925464632930537458693548152239191651779473859039175017703781073285142602369604868181650616917571848228453901192612928304152787673322182240496784499757581974916736593611036624309120739806390671434878884583867516598336146959241170825985297034539611414026394685421970673878433152636150899283675853884618442390476716212388586509960240577957386217016094045307104393405239761705161072313785552978515625
Parter matrix
29:
18389224448555241306916172836519730475708719237866889582517051994821604855284143071510963623865860364253606920477966345122241340475425402259594825652242848598138529505780221746603256820050670621313824029063322601757792250502812188029948327042385162086470151072252660199199473230603160311979630767576491320574714424966595397514757019131185596085209500029380042981833589809239552545024986753882395745714267778373663619222305076694685694308481086630683935244288/253083194744431750124824963121161998285631527170321093653488074466611758928602456830303559040256395678679083828384596981936861185252159400450281876135714764929093480571627990234826375008341877167617676843659598001022023459491512129967865041222541360916283286594933306261471838083055607012298234986262338114402641288932172266203174049426220373493645406944274408099548685328216396511289420408894081899116921940962043890976929105818271636962890625
Parter matrix
30:
11935289041890653422458560211344367566035064229836658432562760370881500666512962196005301087896114215912896307042303231644136800468391898597288380229865050619886197046118267213623392373337061884683299874287304268610214384204772333275145693206369184966882385289582680400610353273504118108261671577527388040961204580193392729480145710139771575499211887880475534583729574257671153630431383455991912285960981537776990485888804447222095975742958525624801339500650274818692551404787439203784851456/52730664990213071403992070529767197055327856846606843602898217457249384424102616905635675434695649060219811896136150228786078171167479019467876065374492199588292762919368386777096731186400959438960695015474123322315277906011099829863289334370987252818231482100559305311308824233046517474879085424846475412911857540180247993242403320611090996514285370920323705683023904719349104829707219198086439965713154061493893868520124670456975483076866861864573365892283618450164794921875
Ris matrix
2:
-4/3
Ris matrix
3:
-256/135
Ris matrix
4:
65536/23625
Ris matrix
5:
1073741824/260465625
Ris matrix
6:
-70368744177664/11371668721875
Ris matrix
7:
-73786976294838206464/7888446990683634375
Ris matrix
8:
309485009821345068724781056/21776965089186101310140625
Ris matrix
9:
332306998946228968225951765070086144/15330043202319289712414934678515625
Ris matrix
10:
-1427247692705959881058285969449495136382746624/43033523436556282747812223470803609794921875
Ris matrix
11:
-98079714615416886934934209737619787751599303819750539264/1927983533652930855481078826533672813447199742802734375
Ris matrix
12:
26959946667150639794667015087019630673637144422540572481103610249216/344795425404576705797704088613314881404393134165837604181728515625
Ris matrix
13:
474284397516047136454946754595585670566993857190463750305618264096412179005177856/3939535380432660481737902361441262572314314558395701458999546507445773681640625
Ris matrix
14:
-33374797436264220037422214158899251790667258161822699530422525122222183215322508594108782608384/179781741661237211642347145472441720566037161335982334498678098832668480910022434539794921875
Ris matrix
15:
-37576681324381331646231689548629392438010920782533117931316655544515344401833735095419183974156299248510959616/131102762724248208694626102055982701077513741180474774872190798192181686391991410348943862541866302490234375
Ris matrix
16:
169230328010303641331690318856389386196071598838855992136870091590247882556495704531248437872567112920983350278405979725889536/381992888274486172330023966308185542489601516762345685811779018531247003918774605768603887469672985625462095737457275390625
Ris matrix
17:
780437137578998057845399307448291576437149535666242787714789239906342934704941405030076525765872992789956732780351655723861993919822071326572544/1138608299507615162647427292330629670698783697607908542468662932849770354828400457047052852766179259956360889577703449255507051944732666015625
Ris matrix
18:
-14396524142538228424993723224595141948383030778566133225922417832357880258148761185020930195532450742879746914027266864394266451377581759004827248578768524336431104/13563680594226352698868417457008363251850061118693602687589871286839911501626127259714664607510903154430496578059712351350009807499277087129704654216766357421875
Ris matrix
19:
-4249103942534136789516705652419749018636744941816255385595553105603228478886817941913300018121834285351114635889972008122772634701221657915276159830132698815550650166683145752253825024/2583244496406594136328099897535836001132395518344940923334796681624438664057934161269921018022327444953972978976807115134633782808793490169963532250695127382553182542324066162109375
Ris matrix
20:
5016456510113118655434598811035278955030765345404790744303017523831112055108147451509157692220295382716162651878526895249385292291816524375083746691371804094271873160484737966720260389217684476157468082176/1966584448933681291700111990511249252996867781069892652927671711453458253490231480360206319125082095026615600393917412850475986162580856819591467906737908092130755099294776354069472290575504302978515625
Ris matrix
21:
379032737378102767370356320425415662904513187772631008578870126471203845870697482014374611530431269030880793627229265919475483409207718357286202948008100864063587640630090308972232735749901964068667724412528434753635948938919936/95756473471527368203359098038636302734320254570510484032372333543233950174955829493086940570565074629250599717033100754347686744918373037370505012065439012316754988621149326723236821614126773416110445396043360233306884765625
Ris matrix
22:
-114555615673899844817675135734699353962886956688682610117765798809244256021411616637381137063298513501717438036875998880297587443028293506801780104166258318482051497231916848409067205045915915104980160032926980989903722214426950334461435148425897181184/18639635230605950582990575619102336720656612652585253503322680860780678791015192710857351144273949000087985951216766601554775840266484387042002198480751269293590050624585780648624718114741661821832371369282735343395152243829215876758098602294921875
Ris matrix
23:
-553956966280111321359151042308621317197106853745652161186848528428353614047320326248246548509656023453846098404449586961587736474553087989908021159880755329796288475560940755137311819879076531853615938045960455092067922915100261601864210866521544040371494407003426519343169536/58023279796528313036778874394104795804150278629297979673062087444214169731311277950493607523991494725719922634357275601790646250234277201850765042424959270476057390617640003320419494644082324993085387906625818657872956700448451188652568700755864483653567731380462646484375
Ris matrix
24:
10715086071862673209484250490600018105614048117055336074437503883703510511249361224931983788156958581275946729175531468251871452856923140435984577574698574803934567774824230985421074605062371141877954182153046474983581941267398767559165543946077062914571196477686542167660429831652624386837205668069376/722140726079450649978046196471342605407045834027578376367384992090511968103289122311552762289829460554916410826200404806737182623725342065933789973823777935755001596224543168482663038365776607387156022411318741208938681865036766463691667048954725551639913210973715109730619587935507297515869140625
Ris matrix
25:
53058536290991634737396540170283858539198978395771271474551549598742698935712572215646062825029533563666321339663466341699370021653261445199826360714064955944866263669556212233526813063143284104557957658305559283253118889734882642786544086063293282737405311454375777285526890640894984855797452638426498665888535738081213096656896/2299814624023306440059372328309540401870462162098782696355188655078330830543381562183319794476727956275051550071401517193197980470453370256239829330997598599886249228660944979630934777555149960305009693579987739018097578513714869135288072761419420858120304547144634856878739999583757563296959602894276031292974948883056640625
Ris matrix
26:
-1050932584007919349247276470706337614964595016201702265730788379907271770705672950402678499587161021318748225038571874254674373003589884095763648064131391439493377897439940849663371575715335395746244327662295930556112407548110386304314732285524778233931590806624721312177403478258221519824050340117043937656005048385392539239027558362141047023676361763979264/29285326003717614607804626715592039854049939876465617875516393067336200504839511603258277578530907832960527472351549373766632209825879529344513468164156788136301696184084726853654890040083335749409558712620378277410078995162362596998825850361411534524766905791872988667930912618485280210592786529340065206396141767506957620526009122841060161590576171875
Ris matrix
27:
-333053830229260770621607690475650737235688391912534553770211595212915457909547529536391662135434373914957491646536992249324274119946350877060477011381674264413572144300455051064741867189170673615134701457526073958774197668721551323742800044302024329335435164329957146513229330720904179349319194620509401659106183110500496672854349935639454656924097627831557980822871062737782501399330816/5964398272540535029373394364875651916634361710058601930960526124478747871090035862673372190515166935687886994684251423287559381642304349511073776331120074272799787226239922242727982724059384648164400666593400970516286320507430210211584256581236940892518782856849779672531137274539963887482758571714650048101771690544028853874785538460366673272311910380949484533630311489105224609375
Ris matrix
28:
422195887798433229774352145185468813016238336131097319380815938242257919877032759055678682325872212299032555117279041959654015279739053454668832857993357497160660916766822054198977340111664599558102697040290856557224005740728229651081256478104685172984831138501872159407593068952851198194574416241549499596609861510631826988369559184497748026624786660912594905737228087742453737599954463816552696563892037688613666816/4857292303245568925464632930537458693548152239191651779473859039175017703781073285142602369604868181650616917571848228453901192612928304152787673322182240496784499757581974916736593611036624309120739806390671434878884583867516598336146959241170825985297034539611414026394685421970673878433152636150899283675853884618442390476716212388586509960240577957386217016094045307104393405239761705161072313785552978515625
Ris matrix
29:
34252599717220740964480066367461775384300796758135575023511521508584925635315744339508868053305634044583154694136169450585248582950654257572432655193363965671767535630204196547350393273030725533727228695644666630323589314711861662622444940809877891254555715382569767905770751145860478863082138388560137377259720908415681491633424571028324954370426945443041702698606233688517571510793616536785654250000170950542508846562924006385143276878597376180224/253083194744431750124824963121161998285631527170321093653488074466611758928602456830303559040256395678679083828384596981936861185252159400450281876135714764929093480571627990234826375008341877167617676843659598001022023459491512129967865041222541360916283286594933306261471838083055607012298234986262338114402641288932172266203174049426220373493645406944274408099548685328216396511289420408894081899116921940962043890976929105818271636962890625
Ris matrix
30:
-11115604119273511155004203516379341078954808628034459829854555773438420767442288059746195644043492354371488380285262345936276764113820994829096254175403202529890646269655103994182676424586271759758982690318770956816351399016354543413171258947224528497906760582311712578507469290405621852969444894723024257422615383000478827841715617235537223051499470957066430322574055062300668685167453676454641191252490075097410460830484002104118443786127982311697061639884742739324040156591161344/52730664990213071403992070529767197055327856846606843602898217457249384424102616905635675434695649060219811896136150228786078171167479019467876065374492199588292762919368386777096731186400959438960695015474123322315277906011099829863289334370987252818231482100559305311308824233046517474879085424846475412911857540180247993242403320611090996514285370920323705683023904719349104829707219198086439965713154061493893868520124670456975483076866861864573365892283618450164794921875
Lotkin matrix
2:
-1/6
Lotkin matrix
3:
1/720
Lotkin matrix
4:
-1/1512000
Lotkin matrix
5:
1/53343360000
Lotkin matrix
6:
-1/31052236723200000
Lotkin matrix
7:
1/295415578275110092800000
Lotkin matrix
8:
-1/45669605890716810734764032000000
Lotkin matrix
9:
1/114309087153174410876339218101043200000000
Lotkin matrix
10:
-1/4620689394791469131629562883903627872698368000000000
Lotkin matrix
11:
1/3011136808823946705032922147912793373280011606588129280000000000
Lotkin matrix
12:
-1/31592214953025376429323789919566365990015723988676538675360358727680000000000
Lotkin matrix
13:
1/5331156872394655932836909965321891579443614660439818237914990159812330245849088000000000000
Lotkin matrix
14:
-1/14458303299170833622669935780693684277310731363637560133092357658898163338323964254612357120000000000000000
Lotkin matrix
15:
1/629796643575644571424873956749132605865041866331920284639689209175668180392761640017274420006539367546880000000000000000000
Lotkin matrix
16:
-1/440407815498535791732812207224368521119815565573058990546270517062213369329462718156914389058051557828293483148324372480000000000000000000000
Lotkin matrix
17:
1/4942027270874967100903058996814368495961513560305783766019299667169445342518766377317787821382384437217218598238642008043467767778639872000000000000000000000000
Lotkin matrix
18:
-1/889620425312832786321065537277250859041703167633117122619673395239780823855201961348839204425730901397422696933743776227539292474221225953143270714376192000000000000000000000000000
Lotkin matrix
19:
1/2568216258292205432690544415297148909001837145941464700877021968653065109969693269003066205999544430344086494067378544016295712073714960989449754469431642550899430045777920000000000000000000000000000000
Lotkin matrix
20:
-1/118872735838426725454582212171380822008770991887674324651659266561720987965532229259379288340828688672028287993363277948588281920985539665169329116207490562051177724458307735890481762889891840000000000000000000000000000000000
Lotkin matrix
21:
1/88199979561883976418086977824695971921879937841892654776972243422809358721553227090111633854510615603921193520868132620213070447188533930280240556432150373429621846498365741339871117226592786853396814637008158720000000000000000000000000000000000000
Lotkin matrix
22:
-1/1048852307710976758384961492923179201921439661953147083271904284056425968067193333268465876596875182485459893221759001689486111390553515191070539294984833258040974217834712485751648749967596565296846913473547898295017841926846659166208000000000000000000000000000000000000000
Lotkin matrix
23:
1/199872949623042424888309701136685403528628679787713138236054719718422589700684870947832275750322893513897864207354849238063750939657114602125983613372993881317172920574345601867182146770221185097600410982608492703852641352738747217972913702493228749905985536000000000000000000000000000000000000000000
Lotkin matrix
24:
-1/610281416575910006549711974276253745740059934698340339907170567299906036586178969793101153491423393325360841902257606335640562682262107713454512025351823264202372772583748010245902753661578061051983951698603299201328515933291719132831192418662005576168021999977132929809012060323840000000000000000000000000000000000000000000000
Lotkin matrix
25:
1/29853300618821674372293960724297271594893189916549109230424847679391102762543568220692157540377662611992399863982199799254557587269179071417311803916222204830486918359631363378301454899053330076774012666643561174259163452997979947100907843560480481157548486628606666788879902190198311186436497583375694954496000000000000000000000000000000000000000000000000
Lotkin matrix
26:
-1/23393554801580195093026097618573496146447871386621631185159270272891118066920627141517042456941649029485289635970295460131416999754835466527960526608437593018327602230536011613281269398370163105341672904713004755502109661898525219338356748773542782567049429375897081408911540209633015113860658717736456417541086812827277969944498248089600000000000000000000000000000000000000000000000000
Lotkin matrix
27:
1/293631302291558141102162076849672261981761044386709633972828472287550241960826427815942298566808801791944051178628984611700420530121653759887008704891365404196623770417256298870492449767746222041792056677904915008344908296993927337720352370734336699340709501860944917917647149550267140223929705314403906093648711641967654884734805100750213267492296239231059425125990400000000000000000000000000000000000000000000000000
Lotkin matrix
28:
-1/59030390063402106547022607675018765835575278840523074632927501627042790874104980381662770864085473975503473695746703544087794180124783846670740430322929008939547521627156630428943986361164893045925750752696813944936194350397696328979080114340890279959145963638445430343231353894079527753907499580745594742510519472359803007693077079108167556566543804166418201777127661489917179239019342084347789312000000000000000000000000000000000000000000000000000
Lotkin matrix
29:
1/190057376078450868423360471321085605789890333443093613805283776908324680309027498253708371515932583477084410505545553799366231212351170601757870704507788594750378864357824559543553819421634817462348167220233651254426547989393800243057607159592270476161518195287477409005141974646339579864154698534866077143891992397146036662685287603699243878847268353071356652962597211265601967483097601783518281315072165340951307978968753032396800000000000000000000000000000000000000000000000000000
Lotkin matrix
30:
-1/9799442700798508168514847165571525542349030323391512634126169543778694582956124180074147455695718108028928360057787036922587934339531178596152471976998362081514355396265244621595090412847397512234797128488293812340564219448038529438586738013800759070671969175069017816874570940481860696092684364044871630566076837832003308025736205139275968661850121952419633258357436942766915456116980723741016392211799929818750026185588760191298772475052556547844260130723987456000000000000000000000000000000000000000000000000000000
Riemann matrix
2:
1
Riemann matrix
3:
4
Riemann matrix
4:
-4
Riemann matrix
5:
96
Riemann matrix
6:
-48
Riemann matrix
7:
-384
Riemann matrix
8:
-3456
Riemann matrix
9:
328320
Riemann matrix
10:
-17280
Riemann matrix
11:
-207360
Riemann matrix
12:
-481697280
Riemann matrix
13:
-516741120
Riemann matrix
14:
79427174400
Riemann matrix
15:
1270834790400
Riemann matrix
16:
681401548800
Riemann matrix
17:
12265227878400
Riemann matrix
18:
-6169334376038400
Riemann matrix
19:
-123386687520768000
Riemann matrix
20:
-158218429759488000
Riemann matrix
21:
47610136717000704000
Riemann matrix
22:
-28967583286591488000
Riemann matrix
23:
-695221998878195712000
Riemann matrix
24:
-17380549971954892800000
Riemann matrix
25:
15059315744060158771200000
Riemann matrix
26:
406601525089624286822400000
Riemann matrix
27:
11384842702509480031027200000
Riemann matrix
28:
25272093761061060398284800000
Riemann matrix
29:
-8083599180907870142595072000000
Riemann matrix
30:
-515844434420335033056755712000000
Definition
For a named classical test matrix $A_n$, one of the standard matrices used to test numerical linear algebra software [7], this table gives the determinant $\det(A_n)$ [1], where $n$ is the order of the matrix.
Parameters
$A_n$
—   matrix family (one of the named test matrix families defined in the comments)
$n$
—   order ($n\geq 1$)
Formulas
(1)
The determinant is the signed constant term of the characteristic polynomial: $\det(A_n)=(-1)^n\chi_{A_n}(0)$. The polynomials $\chi_{A_n}(x)$ for these matrix families are in Characteristic polynomials of the classical test matrices.
(2)
For the Hilbert matrix, $\det(A_n)=c_n^4/c_{2n}$, where $c_m=\prod_{j=1}^{m-1}j!$.
(3)
For the Lehmer matrix, $\det(A_n)=\prod_{j=2}^n(2j-1)/j^2$.
(4)
For the Vandermonde matrix, $\det(A_n)=\prod_{1\leq i<j\leq n}(j-i)$.
(5)
The Cauchy determinant identity $\det(1/(x_i+y_j))=\dfrac{\prod_{i<j}(x_j-x_i)(y_j-y_i)} {\prod_{i,j}(x_i+y_j)}$ applies to the Cauchy matrix with $x_i=i$ and $y_j=j$, to the Parter matrix with $x_i=i$ and $y_j=1/2-j$, and to the Ris matrix with $x_i=-i$ and $y_j=n-j+3/2$ after the common factor $1/2$ is taken from each row.
(6)
For the Fiedler matrix, $\det(A_n)=(-1)^{n-1}2^{n-2}(n-1)$.
(7)
For the Kac-Murdock-Szegő matrix with $\rho=1/2$, $\det(A_n)=(1-\rho^2)^{n-1}=(3/4)^{n-1}$.
(8)
For the Grcar matrix with three superdiagonals, the determinants satisfy $g_1=1$, $g_2=2$, $g_3=4$, $g_4=8$, and $g_n=g_{n-1}+g_{n-2}+g_{n-3}+g_{n-4}$ for $n\geq5$.
(9)
For the Lotkin matrix, $\det(A_n)=(-1)^{n-1}n\det(H_n)$, where $H_n$ is the Hilbert matrix of the same order.
(10)
For the Clement matrix, the determinant is $0$ when $n$ is odd. When $n=2m$, $\det(A_n)=(-1)^m\prod_{j=1}^m(2j-1)^2$.
(11)
The Wilkinson determinants satisfy $d_0=1$, $d_1=(n-1)/2$, and $d_j=a_jd_{j-1}-d_{j-2}$, where $a_j=|(n-1)/2-(j-1)|$ and $d_n=\det(A_n)$.
(12)
The omitted gallery families with their default parameters have $\det(A_n)=1$ for the Pascal, Minij, Frank and Moler matrices, $\det(A_n)=0$ for the Chow matrix when $n>1$, $\det(A_n)=M(n)$ for the Redheffer matrix, where $M$ is the Mertens function [5], and $\det(A_n)=n+1$ for the second-difference matrix.
Comments
(13)
Indices are $1$-based. The matrices are $A_{ij}=1/(i+j-1)$ for the Hilbert matrix [2]; $A_{ij}=\min(i,j)/\max(i,j)$ for the Lehmer matrix [3]; $A_{ij}=i^{j-1}$ for the Vandermonde matrix [6]; and $A_{ij}=1/(i+j)$ for the Cauchy matrix, the scalar default case of gallery("cauchy",n) [7]. The Fiedler matrix is $A_{ij}=|i-j|$. The Kac-Murdock-Szegő matrix is $A_{ij}=\rho^{|i-j|}$ with $\rho=1/2$, that is, $A_{ij}=2^{-|i-j|}$. The Grcar matrix has $-1$ on the subdiagonal and $1$ on the diagonal and on the first three superdiagonals.
(14)
The Wilkinson, Clement, Parter, Ris, Lotkin and Riemann matrices use gallery's default parameters and the definitions in MATLAB gallery documentation [7] or the Test Matrix Toolbox source [8]. The Wilkinson matrix is symmetric tridiagonal with off-diagonal entries $1$ and diagonal entries $|(n-1)/2-(i-1)|$. The Clement matrix is the nonsymmetric tridiagonal matrix with zero diagonal, $A_{i,i-1}=n-i+1$ and $A_{i,i+1}=i$. The Parter matrix is $A_{ij}=1/(i-j+1/2)$.
(15)
The Ris matrix, also called the Dingdong matrix, is $A_{ij}=(1/2)/(n-i-j+3/2)$. The Lotkin matrix is the Hilbert matrix with first row replaced by ones. The Riemann matrix is $B(2:n+1,2:n+1)$ with $B_{ij}=i-1$ if $i\mid j$ and $-1$ otherwise.
(16)
The Pascal matrix [4], Minij, Frank and Moler matrices have determinant $1$ for every order; the Chow matrices with gallery's default parameters are singular for $n>1$; the Redheffer determinants are values of the Mertens function; and the second-difference determinants are $n+1$.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.matrix.constructor import matrix
from sage.rings.rational_field import QQ

# Hilbert matrix, n = 3. The constructor calls the lambda with
# 0-based indices, so i+j+1 here is the 1-based denominator i+j-1.
A = matrix(QQ, 3, 3, lambda i, j: QQ(1) / QQ(i + j + 1))
A.det()
Links
Similar tables
Characteristic polynomials of the classical test matrices —   stores the polynomials whose signed constant terms are the determinants listed here
Eigenvalues of the classical symmetric test matrices —   stores the real eigenvalues whose product is the determinant, for the symmetric families listed here
Condition numbers of the classical test matrices —   stores finite 2-norm condition numbers for nonsingular members of the test matrix families listed here
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (it holds the listed nontrivial determinant families for $2\leq n\leq30$, with only even orders of the Clement matrix because the odd orders are singular; the omitted families have determinants that are identically $1$ or $0$, are Mertens values, or are $n+1$, so they are recorded in (12) rather than as rows)
How they were obtained:

Each matrix is built over $\mathbb{Q}$ and its determinant is computed exactly by fraction-free Bareiss elimination, then checked against the formulas; a direct Leibniz determinant checks every listed family with $n\leq6$. (1) was checked against the characteristic polynomial table on the 138 rows the two tables share, and the omitted small families were checked against (12).



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