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either clarify the topic or aid in computing the terms of the inverse matrix. I found the inclusion of the specific element by element expansion of the 3x3 matrix quite useful. I also strongly disagree with those who claim such a representation is prone to errors. That is, assuming copy and paste is something you are able to handle it is trivial and NOT "asking for making copy errors". I strongly doubt whether either the information about the Cayley-Hamilton decomposition or the information about the representation in terms of 3 column vectors is useful to 99.99% of the readers. I propose the 3x3 section be shortened by removing all the material after the Det(A)=aA+bB+cC AND I also propose abandoning the use of the elements A,B,...,I. They add very little in terms of conciseness; compare Det(A) = a(ei-fh)- b(di-fg)+c(dh-eg) with the above, there is very little space savings but the cost of including 9 more (extraneous) variables is significant in terms of clarity and simplicity. What purpose does it serve?
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application part only talks about few of it. In sub-title least square solutions, the invertible matrix is always used for data analyzing for predicting future data, and people always used it to create model of the relationship between the variables and output. For example, to analyze the price of house, the variables should be area, layout, swimming pool, and place and the output should be price. So, I will add more content on application part.
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Example from the introduction: "Over the field of real numbers, the set of singular n-by-n matrices, considered as a subset of R^{n \times n}, is a null set, i.e., has
Lebesgue measure zero. (This is true because singular matrices can be thought of as the roots of the polynomial function given by the
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I have only a moderate objection to the above cross/dot product representation: it's useless. If what we're looking for is a REPRESENTATION of the inverse of a matrix, then A is hard to beat. I'm writing this 6½ years after
Catskineater's post, but I don't see how an alternative representation helps
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The statement that "It is crucial for the matrix H to be invertible for the receiver to be able to figure out the transmitted information." is just plain wrong. The matrix H is not always square, and there are several better ways of decoding the transmitted signal, instead of inverting the channel
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I really think there should be a mention of linear independence here after the reference to singular matrices since singularity is equivalent to linear dependence, which also ties in to the discussion later about eigenvectors and such. There's already a page on linear independence so just a quick
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I will reorder the theorems to show audiences that they can be explained by each others first. In addition, the formulas are hard to most audiences for understanding, so I will add some explanations on these formulas. Also, Invertible Matrix can be used in many concrete ways in real life but the
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I agree that this article doesn't need an explicit formula for every single dimension, but I still would include a formula for the 3x3 case. Not the one above, which is admittedly long winded & inconvenient, just asking for making copying errors. But there is a much simpler one, only using
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The purpose is to say that singular (non-invertible) matrices are very very very rare. If you choose a matrix with random real entries (say, between 0 and 1), then the probability it is singular is literally zero. That is not to say that non-invertible matrices can't happen, just that they are
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I have referred to
Knowledge's "matrix inverse" page innumerable times in the past to remind me how to do simple matrix inversion. (I forget these things!) Now when I look for that simple information I get overloaded with nonsense about the Lebesque measure of the set of singular matrices.
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Hopefully I copied it over rightly. Looking at the letters like this makes the pattern of 2x2 matrices excluding the row and column of the element in question, turned sideways, seem much more intuitive, although it sure sounds complicated when I say it out like that.
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I'm not 100% sure, but i'm pretty sure that that gives out the solution for the transpose of the inverse, not the inverse itself. A pretty easy and quick fix, but i'm pretty lazy and don't have the time to formulate it into wiki and make sure its all right.
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I think it is important to include that singularity is rare. Maybe we should add something small to make sure we're not overgeneralizing. Singularity over a euclidean field is rare. But isn't a square matrix over z2 almost always singular?
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Who cares? Sure, this may be an interesting aside, but it obfuscates more important things. In particular, since the "matrix inverse" page has been merged here, the extremely useful content that used to reside there should be clearer.
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1177:. That this matrix is a left inverse of A can be checked easily by using basic properties of the cross & triple products. And since left inverses & right inverses are identical for all groups, this is indeed the inverse of A.
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My request to the fantastic math heads writing this page: put the high-school stuff first, since that's what a lot people will want to see. Make it clear to people with minimum math background, and keep it simple when possible.
1453:{\displaystyle \mathbf {A} ^{-1}={\begin{bmatrix}a&b&c\\d&e&f\\g&h&k\\\end{bmatrix}}^{-1}={\frac {1}{Z}}{\begin{bmatrix}\,A&\,B&\,C\\\,D&\,E&\,F\\\,G&\,H&\,K\\\end{bmatrix}}}
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I agree, this page does have a lot of high level terms. Though the concept itself is pretty layered, maybe we should introduce a basic introduction section? I think it would appeal to a wider audience.
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Prompted by the discussion here, I have moved it to later. It does not seem to be important enough to be in the lead of the article. Incidentally, the lead doesn't exactly comply with the
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Thank you for your concern about the minus signs, but
Knowledge has a software bug. Editors have tried several times to kludge it to no avail. This has been reported multiple times now on
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Knowledge used to be a good source for the basic explaination. If I needed more, I'd go to
Mathworld- which was not very often, since I never understood what Wolfram was saying!
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Somehow the formula for the inverse of a general 2x2 matrix is not showing correctly. The latex has a minus before the c but the output does not. I can't seem to fix it.
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infinitely unlikely. (if this seems like a contradiction, consider throwing a dart at a dartboard - what is the probability that the dart will hit a particular point?)
880:{\displaystyle ={\frac {1}{a(ie-hf)-d(ib-hc)+g(fb-ec)}}{\begin{bmatrix}ie-hf&hc-ib&fb-ec\\gf-id&ia-gc&dc-fa\\hd-ge&gb-ha&ea-db\\\end{bmatrix}}}
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inverse. It would be completely unnecessary to show an example of 3x3 in this article, IMO. The only reason that the 2x2 is shown is because it's trivially simple.
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I agree with the above. The formula on the main page caused me lots of problems since it is actually transpose of the inverse, but the above formula seems to work.
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There are links to two different methods for solving systems that involve inverse matrices, as well as a description of the general analytic method for obtaining the
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1874:{\displaystyle {\begin{matrix}A=(ek-fh)&D=(ch-bk)&G=(bf-ce)\\B=(fg-dk)&E=(ak-cg)&H=(cd-af)\\C=(dh-eg)&F=(bg-ah)&K=(ae-bd)\\\end{matrix}}}
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I think this information is important to include, but that earlier in the article we can describe it in simpler terms, such as those used by 67.9.148.47 above.
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I understand why you would not wish to post a general form for the inversion of a 3x3 matrix, but maybe a step by step with a simple example?
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1090:{\displaystyle \mathbf {x_{1}} \times \mathbf {x_{2}} ,\;\mathbf {x_{2}} \times \mathbf {x_{0}} ,\;\mathbf {x_{0}} \times \mathbf {x_{1}} }
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Could someone add a simple example with real numbers? That was what I was looking for, just some actual example not involving variables.
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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matrix. I have never edited a
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is finite (non-zero), the matrix is invertible, with the elements of the above matrix on the right side given by
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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The formula on the page has been corrected more than two years ago, so you are probably misreading something.—
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You guys should probably switch the matrix formula for the inversion of 3x3 matricies on the wiki page.
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601:{\displaystyle A^{-1}={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}}^{-1}}
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A family of vectors are linearly independent if and only if the determinant of their matrix is zero.
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on
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I think those phrases would appear more in text, i.e. someone reading and clicking wants to know
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guideline anyway. It should probably be rewritten and the existing content relocated elsewhere.
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This article was the subject of a Wiki
Education Foundation-supported course assignment, between
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before doing mass systematic removals. This message is updated dynamically through the template
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Thanks for the hard work. I'm just trying to help make the information useful for everyone.
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1170:{\displaystyle \det(A)=\mathbf {x_{0}} \cdot (\mathbf {x_{1}} \times \mathbf {x_{2}} )}
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A square matrix is singular if and only if it's determinant is zero.
2298:. We'll just have to wait until the administrators fix it properly.—
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Knowledge:Redirects for discussion/Log/2023 April 25 § Nonsingular
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Sorry to say so, but I think this page is too complicated.
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for additional information. I made the following changes:
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If an invertible matrix A consists of the column vectors
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1490:
1487:
1484:
1481:
1478:
1475:
1462:
1461:
1460:
1447:
1441:
1436:
1434:
1429:
1427:
1422:
1421:
1418:
1413:
1411:
1406:
1404:
1399:
1398:
1395:
1390:
1388:
1383:
1381:
1376:
1375:
1373:
1366:
1363:
1358:
1353:
1350:
1344:
1338:
1335:
1333:
1330:
1328:
1325:
1324:
1321:
1318:
1316:
1313:
1311:
1308:
1307:
1304:
1301:
1299:
1296:
1294:
1291:
1290:
1288:
1282:
1277:
1274:
1269:
1245:
1244:
1243:
1242:
1241:
1240:
1239:
1238:
1237:
1236:
1224:
1223:
1222:
1221:
1220:
1219:
1218:
1217:
1196:
1195:
1194:
1193:
1192:
1191:
1178:
1166:
1160:
1156:
1151:
1145:
1141:
1136:
1133:
1127:
1123:
1118:
1115:
1112:
1109:
1106:
1083:
1079:
1074:
1068:
1064:
1057:
1051:
1047:
1042:
1036:
1032:
1025:
1019:
1015:
1010:
1004:
1000:
975:
971:
964:
958:
954:
947:
941:
937:
925:
922:triple product
908:
907:
906:
905:
900:72.224.200.135
892:
891:
890:
889:
888:
887:
874:
868:
865:
862:
859:
856:
853:
851:
848:
845:
842:
839:
836:
834:
831:
828:
825:
822:
819:
818:
815:
812:
809:
806:
803:
800:
798:
795:
792:
789:
786:
783:
781:
778:
775:
772:
769:
766:
765:
762:
759:
756:
753:
750:
747:
745:
742:
739:
736:
733:
730:
728:
725:
722:
719:
716:
713:
712:
710:
702:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
623:
618:
608:
595:
592:
586:
580:
577:
575:
572:
570:
567:
566:
563:
560:
558:
555:
553:
550:
549:
546:
543:
541:
538:
536:
533:
532:
530:
524:
519:
516:
512:
496:
495:
494:
493:
483:
482:
453:
450:
449:
448:
433:
432:
431:
430:
429:
428:
427:
416:60.166.111.122
399:
398:
397:
396:
395:
394:
365:
364:
363:
362:
337:
336:
326:
325:
270:
267:
221:
218:
215:
214:
202:
199:
198:
193:
189:
187:
184:
183:
175:
174:
169:
163:
157:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
2401:
2390:
2387:
2385:
2382:
2381:
2379:
2372:
2371:
2368:
2365:
2362:
2358:
2357:
2352:
2348:
2343:
2339:The redirect
2337:
2330:
2326:
2322:
2320:
2313:
2309:
2305:
2301:
2297:
2293:
2292:
2291:
2289:
2285:
2281:
2280:92.194.124.84
2277:
2267:
2265:
2264:
2259:
2254:
2253:
2242:
2238:
2235:
2231:
2230:
2229:
2222:
2216:
2212:
2208:
2204:
2198:
2193:
2188:
2184:
2180:
2179:
2178:
2176:
2172:
2168:
2163:
2157:
2155:
2154:
2150:
2146:
2141:
2139:
2135:
2127:
2123:
2119:
2117:
2116:
2112:
2108:
2104:
2100:
2097:
2091:
2089:
2087:
2083:
2079:
2075:
2064:
2046:
2042:
2038:
2033:
2032:
2031:
2030:
2029:
2028:
2027:
2026:
2025:
2024:
2023:
2022:
2021:
2020:
2019:
2018:
2003:
2000:
1997:
1993:
1992:
1991:
1990:
1989:
1988:
1987:
1986:
1985:
1984:
1983:
1982:
1981:
1980:
1966:
1962:
1958:
1954:
1948:
1947:
1946:
1945:
1944:
1943:
1942:
1941:
1940:
1939:
1938:
1937:
1925:
1921:
1917:
1913:
1909:
1902:
1901:
1900:
1899:
1898:
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1896:
1895:
1894:
1893:
1861:
1858:
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1852:
1849:
1843:
1840:
1832:
1829:
1826:
1823:
1820:
1814:
1811:
1803:
1800:
1797:
1794:
1791:
1785:
1782:
1772:
1769:
1766:
1763:
1760:
1754:
1751:
1743:
1740:
1737:
1734:
1731:
1725:
1722:
1714:
1711:
1708:
1705:
1702:
1696:
1693:
1683:
1680:
1677:
1674:
1671:
1665:
1662:
1654:
1651:
1648:
1645:
1642:
1636:
1633:
1625:
1622:
1619:
1616:
1613:
1607:
1604:
1593:
1592:
1578:
1570:
1551:
1548:
1545:
1542:
1539:
1533:
1530:
1524:
1521:
1518:
1515:
1512:
1506:
1503:
1497:
1494:
1491:
1488:
1485:
1479:
1476:
1473:
1466:
1465:
1463:
1445:
1439:
1432:
1425:
1416:
1409:
1402:
1393:
1386:
1379:
1371:
1364:
1361:
1356:
1351:
1348:
1342:
1336:
1331:
1326:
1319:
1314:
1309:
1302:
1297:
1292:
1286:
1280:
1275:
1272:
1258:
1257:
1255:
1254:
1253:
1252:
1251:
1250:
1249:
1248:
1247:
1246:
1234:
1233:
1232:
1231:
1230:
1229:
1228:
1227:
1226:
1225:
1216:
1212:
1208:
1204:
1203:
1202:
1201:
1200:
1199:
1198:
1197:
1190:
1186:
1182:
1149:
1131:
1116:
1110:
1072:
1055:
1040:
1023:
1008:
962:
945:
923:
919:
918:cross product
914:
913:
912:
911:
910:
909:
904:
901:
896:
895:
894:
893:
872:
866:
863:
860:
857:
854:
849:
846:
843:
840:
837:
832:
829:
826:
823:
820:
813:
810:
807:
804:
801:
796:
793:
790:
787:
784:
779:
776:
773:
770:
767:
760:
757:
754:
751:
748:
743:
740:
737:
734:
731:
726:
723:
720:
717:
714:
708:
697:
694:
691:
688:
685:
679:
676:
670:
667:
664:
661:
658:
652:
649:
643:
640:
637:
634:
631:
625:
621:
616:
609:
593:
590:
584:
578:
573:
568:
561:
556:
551:
544:
539:
534:
528:
522:
517:
514:
510:
502:
501:
500:
499:
498:
497:
491:
487:
486:
485:
484:
481:
478:
474:
470:
466:
465:
464:
462:
461:Nightwindzero
457:
451:
447:
443:
439:
434:
425:
421:
417:
413:
407:
406:
405:
404:
403:
402:
401:
400:
393:
389:
381:
375:
371:
370:
369:
368:
367:
366:
360:
356:
352:
351:132.235.46.80
348:
341:
340:
339:
338:
335:
332:
328:
327:
324:
320:
316:
311:
310:
309:
308:
304:
300:
296:
293:
289:
286:
282:
278:
274:
268:
266:
264:
260:
256:
252:
246:
244:
240:
227:
219:
211:
206:
201:
200:
186:
185:
182:
181:
177:
176:
172:
167:
162:
161:
158:
142:
138:
137:High-priority
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
62:High‑priority
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
2354:
2333:
2327:" listed at
2317:
2274:— Preceding
2271:
2249:
2246:
2221:source check
2200:
2194:
2191:
2164:
2161:
2142:
2137:
2133:
2131:
2125:
2121:
2107:SomeHandyGuy
2105:
2101:
2098:
2095:
2072:— Preceding
2068:
2037:71.30.36.108
1957:65.60.221.79
1951:— Preceding
1906:— Preceding
1207:Catskineater
1181:Catskineater
472:
468:
458:
455:
410:— Preceding
379:siℓℓy rabbit
345:— Preceding
297:
294:
290:
287:
283:
279:
275:
272:
247:
223:
204:
178:
170:
156:
136:
96:
40:WikiProjects
2342:Nonsingular
2325:Nonsingular
315:67.9.148.47
285:Seriously?
112:Mathematics
103:mathematics
59:Mathematics
2378:Categories
2258:Report bug
2300:Anita5192
2241:this tool
2234:this tool
477:Oli Filth
2364:1234qwer
2361:1234qwer
2276:unsigned
2247:Cheers.—
2074:unsigned
1953:unsigned
1920:contribs
1908:unsigned
456:please!
412:unsigned
347:unsigned
331:Dcoetzee
299:Hawkeyek
255:PrimeBOT
205:365 days
171:Archives
2171:my edit
438:SriCHaM
374:WP:LEAD
243:Laoer22
139:on the
30:B-class
2296:WP:VPT
1464:where
920:&
36:scale.
2145:Fmadd
2136:, or
1912:Njc69
488:From
2304:talk
2284:talk
2149:talk
2111:talk
2082:talk
2041:talk
1996:Emil
1961:talk
1916:talk
1256:i.e
1211:talk
1185:talk
442:talk
420:talk
386:talk
355:talk
319:talk
303:talk
269:Sigh
259:talk
233:and
131:High
2215:RfC
2185:to
2124:or
1105:det
253:by
2380::
2306:)
2286:)
2228:.
2223:}}
2219:{{
2151:)
2113:)
2084:)
2043:)
1999:J.
1963:)
1922:)
1918:•
1856:−
1827:−
1798:−
1767:−
1738:−
1709:−
1678:−
1649:−
1620:−
1546:−
1519:−
1492:−
1349:−
1273:−
1213:)
1187:)
1150:×
1132:⋅
1073:×
1041:×
1009:×
924::
861:−
844:−
827:−
808:−
791:−
774:−
755:−
738:−
721:−
692:−
665:−
650:−
638:−
591:−
515:−
444:)
422:)
390:)
357:)
321:)
305:)
261:)
245:.
2367:4
2323:"
2302:(
2282:(
2260:)
2256:(
2243:.
2236:.
2147:(
2128:?
2109:(
2080:(
2039:(
1959:(
1914:(
1865:)
1862:d
1859:b
1853:e
1850:a
1847:(
1844:=
1841:K
1836:)
1833:h
1830:a
1824:g
1821:b
1818:(
1815:=
1812:F
1807:)
1804:g
1801:e
1795:h
1792:d
1789:(
1786:=
1783:C
1776:)
1773:f
1770:a
1764:d
1761:c
1758:(
1755:=
1752:H
1747:)
1744:g
1741:c
1735:k
1732:a
1729:(
1726:=
1723:E
1718:)
1715:k
1712:d
1706:g
1703:f
1700:(
1697:=
1694:B
1687:)
1684:e
1681:c
1675:f
1672:b
1669:(
1666:=
1663:G
1658:)
1655:k
1652:b
1646:h
1643:c
1640:(
1637:=
1634:D
1629:)
1626:h
1623:f
1617:k
1614:e
1611:(
1608:=
1605:A
1579:Z
1555:)
1552:g
1549:e
1543:h
1540:d
1537:(
1534:c
1531:+
1528:)
1525:k
1522:d
1516:g
1513:f
1510:(
1507:b
1504:+
1501:)
1498:h
1495:f
1489:k
1486:e
1483:(
1480:a
1477:=
1474:Z
1446:]
1440:K
1433:H
1426:G
1417:F
1410:E
1403:D
1394:C
1387:B
1380:A
1372:[
1365:Z
1362:1
1357:=
1352:1
1343:]
1337:k
1332:h
1327:g
1320:f
1315:e
1310:d
1303:c
1298:b
1293:a
1287:[
1281:=
1276:1
1268:A
1209:(
1183:(
1165:)
1159:2
1155:x
1144:1
1140:x
1135:(
1126:0
1122:x
1117:=
1114:)
1111:A
1108:(
1082:1
1078:x
1067:0
1063:x
1056:,
1050:0
1046:x
1035:2
1031:x
1024:,
1018:2
1014:x
1003:1
999:x
974:2
970:x
963:,
957:1
953:x
946:,
940:0
936:x
873:]
867:b
864:d
858:a
855:e
850:a
847:h
841:b
838:g
833:e
830:g
824:d
821:h
814:a
811:f
805:c
802:d
797:c
794:g
788:a
785:i
780:d
777:i
771:f
768:g
761:c
758:e
752:b
749:f
744:b
741:i
735:c
732:h
727:f
724:h
718:e
715:i
709:[
701:)
698:c
695:e
689:b
686:f
683:(
680:g
677:+
674:)
671:c
668:h
662:b
659:i
656:(
653:d
647:)
644:f
641:h
635:e
632:i
629:(
626:a
622:1
617:=
594:1
585:]
579:i
574:h
569:g
562:f
557:e
552:d
545:c
540:b
535:a
529:[
523:=
518:1
511:A
492::
473:n
471:x
469:n
440:(
418:(
382:(
353:(
317:(
301:(
257:(
180:1
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
