5536:
5800:
3780:
5154:
2577:
3411:
484:
4925:
3472:
2333:
2042:
2379:
4804:
4525:
312:
3178:
5149:{\displaystyle \mathbf {M} ^{-1}={\frac {1}{\det(M)}}{\begin{bmatrix}A_{11}&A_{21}&\cdots &A_{n1}\\A_{12}&A_{22}&\cdots &A_{n2}\\\vdots &\vdots &\ddots &\vdots \\A_{1n}&A_{2n}&\cdots &A_{nn}\end{bmatrix}}}
2669:
3775:{\displaystyle _{I,J}\det \mathbf {A} (e_{1}\wedge \ldots \wedge e_{n})=\pm (e_{j_{1}})\wedge \ldots \wedge (e_{j_{k}})\wedge (\mathbf {A} e_{i'_{1}})\wedge \ldots \wedge (\mathbf {A} e_{i'_{n-k}})=\pm _{J',I'}(e_{1}\wedge \ldots \wedge e_{n}).}
4682:
287:
2057:
1766:
2572:{\displaystyle \mathbf {C} ={\begin{bmatrix}C_{11}&C_{12}&\cdots &C_{1n}\\C_{21}&C_{22}&\cdots &C_{2n}\\\vdots &\vdots &\ddots &\vdots \\C_{n1}&C_{n2}&\cdots &C_{nn}\end{bmatrix}}}
4394:
1548:
3170:
2954:
1329:
4693:
1074:
3881:
2829:
2758:
969:
898:
4912:
4592:
4405:
585:
479:{\displaystyle M_{2,3}=\det {\begin{bmatrix}1&4&\Box \\\Box &\Box &\Box \\-1&9&\Box \\\end{bmatrix}}=\det {\begin{bmatrix}1&4\\-1&9\\\end{bmatrix}}=9-(-4)=13}
4259:
3406:{\displaystyle _{I,J}(e_{1}\wedge \ldots \wedge e_{n})=\pm (\mathbf {A} ^{-1}e_{j_{1}})\wedge \ldots \wedge (\mathbf {A} ^{-1}e_{j_{k}})\wedge e_{i'_{1}}\wedge \ldots \wedge e_{i'_{n-k}},}
808:
1754:
3460:
2596:
1146:
1110:
1374:
1675:
198:
5350:
4607:
1438:. Which notation is used should always be checked from the source in question. In this article, we use the inclusive definition of choosing the elements from rows of
1218:
3034:
1185:
4100:
for the determinant of the product of two matrices are special cases of the following general statement about the minors of a product of two matrices. Suppose that
209:
2328:{\displaystyle \ \det(\mathbf {A} )=a_{i1}C_{i1}+a_{i2}C_{i2}+a_{i3}C_{i3}+\cdots +a_{in}C_{in}=\sum _{j=1}^{n}a_{ij}C_{ij}=\sum _{j=1}^{n}a_{ij}(-1)^{i+j}M_{ij}}
2037:{\displaystyle \ \det(\mathbf {A} )=a_{1j}C_{1j}+a_{2j}C_{2j}+a_{3j}C_{3j}+\cdots +a_{nj}C_{nj}=\sum _{i=1}^{n}a_{ij}C_{ij}=\sum _{i=1}^{n}a_{ij}(-1)^{i+j}M_{ij}}
1400:
1414:, some authors mean the determinant of the matrix that is formed as above, by taking the elements of the original matrix from the rows whose indexes are in
1685:), can be written as the sum of the cofactors of any row or column of the matrix multiplied by the entries that generated them. In other words, defining
1406:, etc.), depending on the source. Also, there are two types of denotations in use in literature: by the minor associated to ordered sequences of indexes
5394:
4329:
3051:
1457:
5785:
5315:
5269:
5231:
1223:
4799:{\displaystyle -13\mathbf {e} _{1}\wedge \mathbf {e} _{2}-7\mathbf {e} _{1}\wedge \mathbf {e} _{3}+5\mathbf {e} _{2}\wedge \mathbf {e} _{3}}
971:
be ordered sequences (in natural order, as it is always assumed when talking about minors unless otherwise stated) of indexes, call them
2849:
982:
3788:
5291:
4530:
where the two expressions correspond to the two columns of our matrix. Using the properties of the wedge product, namely that it is
4399:
are −13 (from the first two rows), −7 (from the first and last row), and 5 (from the last two rows). Now consider the wedge product
2763:
2692:
903:
832:
76:
of square matrices. The requirement that the square matrix be smaller than the original matrix is often omitted in the definition.
4842:
4520:{\displaystyle (\mathbf {e} _{1}+3\mathbf {e} _{2}+2\mathbf {e} _{3})\wedge (4\mathbf {e} _{1}-\mathbf {e} _{2}+\mathbf {e} _{3})}
5775:
4544:
5737:
5673:
64:
by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (
678:(the word "determinant" is often omitted, and the word "degree" is sometimes used instead of "order") is the determinant of a
5337:(1st ed., original language is Russian), Moscow: State Publishing House of technical and theoretical literature, 1953, p.491,
1625:
for the expansion of determinants, which is a method of computing larger determinants in terms of smaller ones. Given an
5515:
5387:
4081:
495:
5620:
5470:
4535:
4158:
5525:
5419:
2967:
denote the ordered sequences of indices (the indices are in natural order of magnitude, as above) complementary to
749:
5765:
5414:
5640:
4097:
5757:
5264:
Linear
Algebra and Geometry, Igor R. Shafarevich, Alexey O. Remizov, Springer-Verlag Berlin Heidelberg, 2013,
4077:
4085:
1688:
5803:
5510:
5380:
2664:{\displaystyle \mathbf {A} ^{-1}={\frac {1}{\operatorname {det} (\mathbf {A} )}}\mathbf {C} ^{\mathsf {T}}.}
5824:
5500:
5490:
5249:
3419:
5829:
5727:
5582:
5577:
5572:
5505:
5450:
4093:
4069:
of a matrix is the determinant of a square submatrix that is of maximal size with nonzero determinant.
5592:
5557:
5544:
5435:
4035:
3916:
46:
5770:
5650:
5625:
5475:
4285:
3912:
30:
This article is about a concept in linear algebra. For the concept of "minor" in graph theory, see
17:
1115:
1079:
5480:
4677:{\displaystyle \mathbf {e} _{i}\wedge \mathbf {e} _{j}=-\mathbf {e} _{j}\wedge \mathbf {e} _{i},}
1334:
1638:
161:
5678:
5635:
5562:
5455:
5311:
5287:
5281:
5265:
5227:
5210:
4598:
2350:
2344:
1622:
1616:
5305:
1190:
282:{\displaystyle {\begin{bmatrix}1&4&7\\3&0&5\\-1&9&11\\\end{bmatrix}}}
5683:
5587:
5440:
5330:
5172:
4073:
2998:
2354:
1151:
5171:. In modern terminology, the "adjoint" of a matrix most often refers to the corresponding
5742:
5535:
5495:
5485:
5189:
4038:
of the larger matrix (i.e., it consists of matrix elements in rows and columns from 1 to
1379:
5747:
5732:
5668:
5403:
5226:
Elementary Matrix
Algebra (Third edition), Franz E. Hohn, The Macmillan Company, 1973,
4301:
73:
38:
1430:
the determinant of the matrix formed from the original matrix by deleting the rows in
5818:
5780:
5703:
5663:
5630:
5610:
4289:
4042:, also known as a leading principal submatrix), then the principal minor is called a
57:
5364:
5356:
742:, leaving the term "minor" to refer to the determinant of this matrix. For a matrix
5713:
5602:
5552:
5445:
4531:
4276:
elements. This formula is a straightforward extension of the Cauchy–Binet formula.
2586:
is the transpose of the cofactor matrix times the reciprocal of the determinant of
5693:
5658:
5615:
5460:
5211:
Theory of
Equations: with an Introduction to the Theory of Binary Algebraic Form
3908:
122:
53:
31:
306:, we find the determinant of the above matrix with row 2 and column 3 removed.
5722:
5465:
4389:{\displaystyle {\begin{pmatrix}1&4\\3&\!\!-1\\2&1\\\end{pmatrix}}}
1543:{\textstyle M_{i,j}=\det \left(\left(A_{p,q}\right)_{p\neq i,q\neq j}\right)}
5520:
5184:
3165:{\displaystyle _{I,J}=\det \left((A_{i_{p},j_{q}})_{p,q=1,\ldots ,k}\right)}
126:
4034:
If the matrix that corresponds to a principal minor is a square upper-left
2357:, as follows. The matrix formed by all of the cofactors of a square matrix
1550:
is standard everywhere in the literature and is used in this article also.
5688:
5164:
2675:
5168:
1324:{\displaystyle M_{i_{1},i_{2},\ldots ,i_{k},j_{1},j_{2},\ldots ,j_{k}}}
612:
203:
To illustrate these definitions, consider the following 3 by 3 matrix,
5698:
4129:
3950:
1595:
have been removed. The complement of the first minor of an element
4323:-vectors. For example, the 2 × 2 minors of the matrix
1069:{\textstyle \det \left((A_{i_{p},j_{q}})_{p,q=1,\ldots ,k}\right)}
72:, which in turn are useful for computing both the determinant and
5372:
2949:{\displaystyle _{I,J}=\pm {\frac {_{J',I'}}{\det \mathbf {A} }},}
3876:{\displaystyle (-1)^{\sum _{s=1}^{k}i_{s}-\sum _{s=1}^{k}j_{s}}}
5376:
4809:
where the coefficients agree with the minors computed earlier.
2824:{\displaystyle 1\leq j_{1}<j_{2}<\ldots <j_{k}\leq n}
2753:{\displaystyle 1\leq i_{1}<i_{2}<\ldots <i_{k}\leq n}
1446:. The exceptional case is the case of the first minor or the (
964:{\displaystyle 1\leq j_{1}<j_{2}<\cdots <j_{k}\leq n}
893:{\displaystyle 1\leq i_{1}<i_{2}<\cdots <i_{k}\leq m}
5243:
5241:
5239:
4907:{\displaystyle \mathbf {A} _{ij}=(-1)^{i+j}\mathbf {M} _{ij}}
4284:
A more systematic, algebraic treatment of minors is given in
1454:)-minor described above; in that case, the exclusive meaning
4919:
Using this notation the inverse matrix is written this way:
5208:
Burnside, William Snow & Panton, Arthur
William (1886)
3172:. A simple proof can be given using wedge product. Indeed,
1422:, whereas some other authors mean by a minor associated to
4587:{\displaystyle \mathbf {e} _{i}\wedge \mathbf {e} _{i}=0,}
2831:
be ordered sequences (in natural order) of indexes (here
4076:, the leading principal minors can be used to test for
3883:, so the sign is determined by the sums of elements in
4973:
4338:
2396:
1460:
985:
752:
415:
343:
218:
4928:
4845:
4696:
4610:
4547:
4408:
4332:
4161:
3791:
3475:
3422:
3181:
3054:
3001:
2852:
2766:
2695:
2689:
The above formula can be generalized as follows: Let
2599:
2382:
2060:
1769:
1691:
1641:
1382:
1337:
1226:
1193:
1154:
1118:
1082:
1076:
corresponding to these choices of indexes is denoted
906:
835:
734:
columns), but this matrix should be referred to as a
498:
315:
212:
164:
706:
columns. Sometimes the term is used to refer to the
5756:
5712:
5649:
5601:
5543:
5428:
2674:The transpose of the cofactor matrix is called the
822:is often defined to be 1. For a square matrix, the
5148:
4906:
4798:
4676:
4586:
4519:
4388:
4253:
3933:We will use the following notation for minors: if
3875:
3774:
3454:
3405:
3164:
3028:
2948:
2823:
2752:
2663:
2571:
2327:
2036:
1748:
1669:
1542:
1394:
1368:
1323:
1212:
1179:
1140:
1104:
1068:
963:
892:
802:
580:{\displaystyle \ C_{2,3}=(-1)^{2+3}(M_{2,3})=-13.}
579:
478:
281:
192:
4359:
4358:
4319:minors appear as the components of the resulting
4080:and the principal minors can be used to test for
4953:
3510:
3082:
2932:
2064:
1773:
1480:
1119:
1084:
986:
407:
335:
5365:Springer Encyclopedia of Mathematics entry for
4307:If the columns of a matrix are wedged together
137: th column. This number is often denoted
4254:{\displaystyle _{I,J}=\sum _{K}_{I,K}_{K,J}\,}
27:Determinant of a subsection of a square matrix
5388:
3040:formed by choosing the rows of the index set
1576:, is formed by the determinant of the matrix
803:{\textstyle {m \choose k}\cdot {n \choose k}}
794:
781:
769:
756:
8:
5304:Viktor Vasil_evich Prasolov (13 June 1994).
3036:denotes the determinant of the submatrix of
5310:. American Mathematical Soc. pp. 15–.
4296:-minors of a matrix are the entries in the
3997:that corresponds to the rows with index in
5395:
5381:
5373:
5286:, p.135, Cambridge University Press, 1999
3922:, then there exists at least one non-zero
5129:
5109:
5094:
5055:
5038:
5026:
5009:
4992:
4980:
4968:
4947:
4935:
4930:
4927:
4895:
4890:
4877:
4852:
4847:
4844:
4834:and defined in the same way as cofactor:
4790:
4785:
4775:
4770:
4757:
4752:
4742:
4737:
4724:
4719:
4709:
4704:
4695:
4665:
4660:
4650:
4645:
4632:
4627:
4617:
4612:
4609:
4569:
4564:
4554:
4549:
4546:
4508:
4503:
4493:
4488:
4478:
4473:
4454:
4449:
4436:
4431:
4418:
4413:
4407:
4333:
4331:
4250:
4238:
4229:
4214:
4205:
4196:
4177:
4165:
4160:
3930:minor, while all larger minors are zero.
3865:
3855:
3844:
3831:
3821:
3810:
3805:
3790:
3760:
3741:
3712:
3703:
3674:
3669:
3660:
3634:
3629:
3620:
3603:
3598:
3571:
3566:
3544:
3525:
3513:
3498:
3485:
3480:
3474:
3446:
3427:
3421:
3383:
3378:
3354:
3349:
3331:
3326:
3313:
3308:
3284:
3279:
3266:
3261:
3242:
3223:
3204:
3191:
3186:
3180:
3127:
3115:
3102:
3097:
3067:
3058:
3053:
3014:
3005:
3000:
2935:
2908:
2899:
2893:
2875:
2862:
2857:
2851:
2809:
2790:
2777:
2765:
2738:
2719:
2706:
2694:
2651:
2650:
2645:
2633:
2618:
2606:
2601:
2598:
2552:
2532:
2517:
2478:
2461:
2449:
2432:
2415:
2403:
2391:
2383:
2381:
2316:
2300:
2278:
2268:
2257:
2241:
2228:
2218:
2207:
2191:
2178:
2156:
2143:
2127:
2114:
2098:
2085:
2070:
2059:
2025:
2009:
1987:
1977:
1966:
1950:
1937:
1927:
1916:
1900:
1887:
1865:
1852:
1836:
1823:
1807:
1794:
1779:
1768:
1737:
1721:
1696:
1690:
1655:
1640:
1512:
1496:
1465:
1459:
1381:
1342:
1336:
1313:
1294:
1281:
1268:
1249:
1236:
1231:
1225:
1198:
1192:
1165:
1153:
1126:
1117:
1087:
1081:
1031:
1019:
1006:
1001:
984:
949:
930:
917:
905:
878:
859:
846:
834:
793:
780:
778:
768:
755:
753:
751:
553:
534:
506:
497:
410:
338:
320:
314:
213:
211:
178:
163:
5353:at Google Video, from MIT OpenCourseWare
158:is obtained by multiplying the minor by
5351:MIT Linear Algebra Lecture on Cofactors
5307:Problems and Theorems in Linear Algebra
5201:
4264:where the sum extends over all subsets
1749:{\displaystyle C_{ij}=(-1)^{i+j}M_{ij}}
826:is just the determinant of the matrix.
5786:Comparison of linear algebra libraries
5222:
5220:
2987:, but not in both (similarly for the
2652:
1756:then the cofactor expansion along the
489:So the cofactor of the (2,3) entry is
68:) are required for calculating matrix
4048:corner (principal) minor (of order k)
2349:One can write down the inverse of an
1621:The cofactors feature prominently in
1611:Cofactor expansion of the determinant
7:
4825:is used. Moreover, it is denoted as
4044:leading principal minor (of order k)
2353:by computing its cofactors by using
1606:Applications of minors and cofactors
4687:we can simplify this expression to
3911:entries (or entries from any other
3455:{\displaystyle e_{1},\ldots ,e_{n}}
105: th column (also called the (
18:User:Cronholm144/Cofactor expansion
785:
760:
25:
4813:A remark about different notation
3785:The sign can be worked out to be
3462:are the basis vectors. Acting by
2047:The cofactor expansion along the
1418:and columns whose indexes are in
5799:
5798:
5776:Basic Linear Algebra Subprograms
5534:
4931:
4891:
4848:
4786:
4771:
4753:
4738:
4720:
4705:
4661:
4646:
4628:
4613:
4565:
4550:
4504:
4489:
4474:
4450:
4432:
4414:
4230:
4206:
4169:
4166:
3704:
3661:
3621:
3514:
3481:
3309:
3262:
3187:
3059:
3006:
2936:
2900:
2858:
2646:
2634:
2602:
2384:
2071:
1780:
1402:denotes the sequence of indexes
5674:Seven-dimensional cross product
5283:Methods of Mathematical Physics
2979:appears exactly once in either
746:as above, there are a total of
4962:
4956:
4874:
4864:
4514:
4466:
4460:
4409:
4235:
4226:
4211:
4202:
4174:
4162:
4092:Both the formula for ordinary
4001:and the columns with index in
3802:
3792:
3766:
3734:
3709:
3700:
3691:
3657:
3645:
3617:
3611:
3591:
3579:
3559:
3550:
3518:
3495:
3476:
3339:
3304:
3292:
3257:
3248:
3216:
3201:
3182:
3124:
3090:
3064:
3055:
3011:
3002:
2975:, so that every index 1, ...,
2905:
2896:
2872:
2853:
2638:
2630:
2297:
2287:
2075:
2067:
2006:
1996:
1784:
1776:
1718:
1708:
1664:
1648:
1389:
1383:
1361:
1355:
1349:
1343:
1162:
1155:
1028:
994:
565:
546:
531:
521:
467:
458:
175:
165:
1:
93:is a square matrix, then the
5516:Eigenvalues and eigenvectors
4280:Multilinear algebra approach
1141:{\displaystyle \det A_{I,J}}
1105:{\displaystyle \det _{I,J}A}
648:minor determinant of order k
5251:Encyclopedia of Mathematics
1369:{\displaystyle M_{(i),(j)}}
80:Definition and illustration
5846:
4817:In some books, instead of
2342:
1670:{\displaystyle A=(a_{ij})}
1614:
979:, respectively. The minor
193:{\displaystyle (-1)^{i+j}}
29:
5794:
5532:
5410:
4082:positive semidefiniteness
4062:leading principal minors.
4058:square matrix, there are
3044:and columns of index set
1580:from which all the rows (
3973:elements, then we write
3466:on both sides, one gets
2678:matrix (also called the
1760: th column gives:
1602:is merely that element.
1213:{\displaystyle M_{I,J}}
129:formed by deleting the
5501:Row and column vectors
5150:
4908:
4800:
4678:
4588:
4521:
4390:
4255:
4144:is a subset of {1,...,
3965:is a subset of {1,...,
3877:
3860:
3826:
3776:
3456:
3407:
3166:
3030:
3029:{\displaystyle _{I,J}}
2950:
2825:
2754:
2665:
2573:
2329:
2273:
2223:
2038:
1982:
1932:
1750:
1671:
1572:, of a square matrix,
1544:
1396:
1370:
1325:
1214:
1181:
1180:{\displaystyle _{I,J}}
1142:
1106:
1070:
965:
894:
804:
718:as above (by deleting
581:
480:
283:
194:
5506:Row and column spaces
5451:Scalar multiplication
5151:
4909:
4801:
4679:
4589:
4522:
4391:
4256:
4094:matrix multiplication
4086:Sylvester's criterion
4078:positive definiteness
3878:
3840:
3806:
3777:
3457:
3408:
3167:
3031:
2951:
2826:
2755:
2666:
2574:
2330:
2253:
2203:
2051: th row gives:
2039:
1962:
1912:
1751:
1677:, the determinant of
1672:
1545:
1397:
1371:
1326:
1215:
1182:
1143:
1107:
1071:
966:
895:
805:
714:matrix obtained from
686:matrix obtained from
582:
481:
292:To compute the minor
284:
195:
5641:Gram–Schmidt process
5593:Gaussian elimination
5357:PlanetMath entry of
4926:
4843:
4694:
4608:
4545:
4406:
4330:
4159:
4098:Cauchy–Binet formula
3789:
3473:
3420:
3179:
3052:
2999:
2850:
2764:
2693:
2597:
2582:Then the inverse of
2380:
2058:
1767:
1689:
1639:
1458:
1380:
1335:
1224:
1191:
1152:
1116:
1080:
983:
904:
833:
750:
672:th minor determinant
496:
313:
210:
162:
97:of the entry in the
5771:Numerical stability
5651:Multilinear algebra
5626:Inner product space
5476:Linear independence
4286:multilinear algebra
3688:
3642:
3397:
3362:
2367:matrix of cofactors
2339:Inverse of a matrix
1395:{\displaystyle (i)}
820:minor of order zero
5481:Linear combination
5335:Theory of matrices
5159:Keep in mind that
5146:
5140:
4904:
4796:
4674:
4584:
4517:
4386:
4380:
4251:
4201:
4074:Hermitian matrices
3895:Other applications
3873:
3772:
3670:
3630:
3452:
3403:
3379:
3350:
3162:
3026:
2946:
2821:
2750:
2661:
2569:
2563:
2325:
2034:
1746:
1667:
1588:) associated with
1540:
1392:
1366:
1321:
1210:
1177:
1138:
1102:
1098:
1066:
961:
890:
800:
736:(square) submatrix
591:General definition
577:
476:
443:
398:
279:
273:
190:
133: th row and
101: th row and
5812:
5811:
5679:Geometric algebra
5636:Kronecker product
5471:Linear projection
5456:Vector projection
5317:978-0-8218-0236-6
5280:Bertha Jeffreys,
5270:978-3-642-30993-9
5232:978-0-02-355950-1
4966:
4192:
4088:for more details.
2941:
2680:classical adjoint
2642:
2365:(also called the
2351:invertible matrix
2345:Invertible matrix
2063:
1772:
1623:Laplace's formula
1617:Laplace expansion
1083:
792:
767:
501:
299:and the cofactor
16:(Redirected from
5837:
5802:
5801:
5684:Exterior algebra
5621:Hadamard product
5538:
5526:Linear equations
5397:
5390:
5383:
5374:
5338:
5331:Felix Gantmacher
5328:
5322:
5321:
5301:
5295:
5278:
5272:
5262:
5256:
5255:
5245:
5234:
5224:
5215:
5206:
5173:adjoint operator
5155:
5153:
5152:
5147:
5145:
5144:
5137:
5136:
5117:
5116:
5102:
5101:
5063:
5062:
5043:
5042:
5031:
5030:
5017:
5016:
4997:
4996:
4985:
4984:
4967:
4965:
4948:
4943:
4942:
4934:
4913:
4911:
4910:
4905:
4903:
4902:
4894:
4888:
4887:
4860:
4859:
4851:
4805:
4803:
4802:
4797:
4795:
4794:
4789:
4780:
4779:
4774:
4762:
4761:
4756:
4747:
4746:
4741:
4729:
4728:
4723:
4714:
4713:
4708:
4683:
4681:
4680:
4675:
4670:
4669:
4664:
4655:
4654:
4649:
4637:
4636:
4631:
4622:
4621:
4616:
4593:
4591:
4590:
4585:
4574:
4573:
4568:
4559:
4558:
4553:
4526:
4524:
4523:
4518:
4513:
4512:
4507:
4498:
4497:
4492:
4483:
4482:
4477:
4459:
4458:
4453:
4441:
4440:
4435:
4423:
4422:
4417:
4395:
4393:
4392:
4387:
4385:
4384:
4260:
4258:
4257:
4252:
4249:
4248:
4233:
4225:
4224:
4209:
4200:
4188:
4187:
4172:
3992:
3882:
3880:
3879:
3874:
3872:
3871:
3870:
3869:
3859:
3854:
3836:
3835:
3825:
3820:
3781:
3779:
3778:
3773:
3765:
3764:
3746:
3745:
3733:
3732:
3731:
3720:
3707:
3690:
3689:
3684:
3664:
3644:
3643:
3638:
3624:
3610:
3609:
3608:
3607:
3578:
3577:
3576:
3575:
3549:
3548:
3530:
3529:
3517:
3509:
3508:
3493:
3492:
3484:
3461:
3459:
3458:
3453:
3451:
3450:
3432:
3431:
3412:
3410:
3409:
3404:
3399:
3398:
3393:
3364:
3363:
3358:
3338:
3337:
3336:
3335:
3321:
3320:
3312:
3291:
3290:
3289:
3288:
3274:
3273:
3265:
3247:
3246:
3228:
3227:
3215:
3214:
3199:
3198:
3190:
3171:
3169:
3168:
3163:
3161:
3157:
3156:
3155:
3122:
3121:
3120:
3119:
3107:
3106:
3078:
3077:
3062:
3035:
3033:
3032:
3027:
3025:
3024:
3009:
2955:
2953:
2952:
2947:
2942:
2940:
2939:
2930:
2929:
2928:
2927:
2916:
2903:
2894:
2886:
2885:
2870:
2869:
2861:
2830:
2828:
2827:
2822:
2814:
2813:
2795:
2794:
2782:
2781:
2759:
2757:
2756:
2751:
2743:
2742:
2724:
2723:
2711:
2710:
2670:
2668:
2667:
2662:
2657:
2656:
2655:
2649:
2643:
2641:
2637:
2619:
2614:
2613:
2605:
2578:
2576:
2575:
2570:
2568:
2567:
2560:
2559:
2540:
2539:
2525:
2524:
2486:
2485:
2466:
2465:
2454:
2453:
2440:
2439:
2420:
2419:
2408:
2407:
2387:
2334:
2332:
2331:
2326:
2324:
2323:
2311:
2310:
2286:
2285:
2272:
2267:
2249:
2248:
2236:
2235:
2222:
2217:
2199:
2198:
2186:
2185:
2164:
2163:
2151:
2150:
2135:
2134:
2122:
2121:
2106:
2105:
2093:
2092:
2074:
2061:
2043:
2041:
2040:
2035:
2033:
2032:
2020:
2019:
1995:
1994:
1981:
1976:
1958:
1957:
1945:
1944:
1931:
1926:
1908:
1907:
1895:
1894:
1873:
1872:
1860:
1859:
1844:
1843:
1831:
1830:
1815:
1814:
1802:
1801:
1783:
1770:
1755:
1753:
1752:
1747:
1745:
1744:
1732:
1731:
1704:
1703:
1676:
1674:
1673:
1668:
1663:
1662:
1634:
1558:The complement,
1549:
1547:
1546:
1541:
1539:
1535:
1534:
1511:
1507:
1506:
1476:
1475:
1401:
1399:
1398:
1393:
1375:
1373:
1372:
1367:
1365:
1364:
1330:
1328:
1327:
1322:
1320:
1319:
1318:
1317:
1299:
1298:
1286:
1285:
1273:
1272:
1254:
1253:
1241:
1240:
1219:
1217:
1216:
1211:
1209:
1208:
1186:
1184:
1183:
1178:
1176:
1175:
1147:
1145:
1144:
1139:
1137:
1136:
1111:
1109:
1108:
1103:
1097:
1075:
1073:
1072:
1067:
1065:
1061:
1060:
1059:
1026:
1025:
1024:
1023:
1011:
1010:
970:
968:
967:
962:
954:
953:
935:
934:
922:
921:
899:
897:
896:
891:
883:
882:
864:
863:
851:
850:
809:
807:
806:
801:
799:
798:
797:
784:
774:
773:
772:
759:
586:
584:
583:
578:
564:
563:
545:
544:
517:
516:
499:
485:
483:
482:
477:
448:
447:
403:
402:
331:
330:
288:
286:
285:
280:
278:
277:
199:
197:
196:
191:
189:
188:
60:, cut down from
56:of some smaller
21:
5845:
5844:
5840:
5839:
5838:
5836:
5835:
5834:
5815:
5814:
5813:
5808:
5790:
5752:
5708:
5645:
5597:
5539:
5530:
5496:Change of basis
5486:Multilinear map
5424:
5406:
5401:
5347:
5342:
5341:
5329:
5325:
5318:
5303:
5302:
5298:
5279:
5275:
5263:
5259:
5247:
5246:
5237:
5225:
5218:
5207:
5203:
5198:
5190:Compound matrix
5181:
5139:
5138:
5125:
5123:
5118:
5105:
5103:
5090:
5087:
5086:
5081:
5076:
5071:
5065:
5064:
5051:
5049:
5044:
5034:
5032:
5022:
5019:
5018:
5005:
5003:
4998:
4988:
4986:
4976:
4969:
4952:
4929:
4924:
4923:
4889:
4873:
4846:
4841:
4840:
4833:
4815:
4784:
4769:
4751:
4736:
4718:
4703:
4692:
4691:
4659:
4644:
4626:
4611:
4606:
4605:
4563:
4548:
4543:
4542:
4502:
4487:
4472:
4448:
4430:
4412:
4404:
4403:
4379:
4378:
4373:
4367:
4366:
4356:
4350:
4349:
4344:
4334:
4328:
4327:
4315: ×
4311:at a time, the
4282:
4234:
4210:
4173:
4157:
4156:
4152:elements. Then
4120: ×
4108: ×
4054: ×
4029:principal minor
4026:
3988: ×
3984:
3982:
3941: ×
3926: ×
3903: ×
3897:
3861:
3827:
3801:
3787:
3786:
3756:
3737:
3724:
3713:
3708:
3665:
3625:
3599:
3594:
3567:
3562:
3540:
3521:
3494:
3479:
3471:
3470:
3442:
3423:
3418:
3417:
3374:
3345:
3327:
3322:
3307:
3280:
3275:
3260:
3238:
3219:
3200:
3185:
3177:
3176:
3123:
3111:
3098:
3093:
3089:
3085:
3063:
3050:
3049:
3010:
2997:
2996:
2931:
2920:
2909:
2904:
2895:
2871:
2856:
2848:
2847:
2839: ×
2805:
2786:
2773:
2762:
2761:
2734:
2715:
2702:
2691:
2690:
2644:
2623:
2600:
2595:
2594:
2562:
2561:
2548:
2546:
2541:
2528:
2526:
2513:
2510:
2509:
2504:
2499:
2494:
2488:
2487:
2474:
2472:
2467:
2457:
2455:
2445:
2442:
2441:
2428:
2426:
2421:
2411:
2409:
2399:
2392:
2378:
2377:
2369:or, sometimes,
2363:cofactor matrix
2347:
2341:
2312:
2296:
2274:
2237:
2224:
2187:
2174:
2152:
2139:
2123:
2110:
2094:
2081:
2056:
2055:
2021:
2005:
1983:
1946:
1933:
1896:
1883:
1861:
1848:
1832:
1819:
1803:
1790:
1765:
1764:
1733:
1717:
1692:
1687:
1686:
1651:
1637:
1636:
1630: ×
1626:
1619:
1613:
1608:
1600:
1593:
1584:) and columns (
1570:
1563:
1556:
1492:
1488:
1487:
1483:
1461:
1456:
1455:
1442:and columns of
1434:and columns in
1378:
1377:
1338:
1333:
1332:
1309:
1290:
1277:
1264:
1245:
1232:
1227:
1222:
1221:
1194:
1189:
1188:
1161:
1150:
1149:
1122:
1114:
1113:
1078:
1077:
1027:
1015:
1002:
997:
993:
989:
981:
980:
945:
926:
913:
902:
901:
874:
855:
842:
831:
830:
814: ×
810:minors of size
779:
754:
748:
747:
710: ×
682: ×
635: ×
603: ×
593:
549:
530:
502:
494:
493:
442:
441:
436:
427:
426:
421:
411:
397:
396:
391:
386:
377:
376:
371:
366:
360:
359:
354:
349:
339:
316:
311:
310:
305:
298:
272:
271:
266:
261:
252:
251:
246:
241:
235:
234:
229:
224:
214:
208:
207:
174:
160:
159:
145:
87:
82:
35:
28:
23:
22:
15:
12:
11:
5:
5843:
5841:
5833:
5832:
5827:
5817:
5816:
5810:
5809:
5807:
5806:
5795:
5792:
5791:
5789:
5788:
5783:
5778:
5773:
5768:
5766:Floating-point
5762:
5760:
5754:
5753:
5751:
5750:
5748:Tensor product
5745:
5740:
5735:
5733:Function space
5730:
5725:
5719:
5717:
5710:
5709:
5707:
5706:
5701:
5696:
5691:
5686:
5681:
5676:
5671:
5669:Triple product
5666:
5661:
5655:
5653:
5647:
5646:
5644:
5643:
5638:
5633:
5628:
5623:
5618:
5613:
5607:
5605:
5599:
5598:
5596:
5595:
5590:
5585:
5583:Transformation
5580:
5575:
5573:Multiplication
5570:
5565:
5560:
5555:
5549:
5547:
5541:
5540:
5533:
5531:
5529:
5528:
5523:
5518:
5513:
5508:
5503:
5498:
5493:
5488:
5483:
5478:
5473:
5468:
5463:
5458:
5453:
5448:
5443:
5438:
5432:
5430:
5429:Basic concepts
5426:
5425:
5423:
5422:
5417:
5411:
5408:
5407:
5404:Linear algebra
5402:
5400:
5399:
5392:
5385:
5377:
5371:
5370:
5362:
5354:
5346:
5345:External links
5343:
5340:
5339:
5323:
5316:
5296:
5273:
5257:
5235:
5216:
5200:
5199:
5197:
5194:
5193:
5192:
5187:
5180:
5177:
5157:
5156:
5143:
5135:
5132:
5128:
5124:
5122:
5119:
5115:
5112:
5108:
5104:
5100:
5097:
5093:
5089:
5088:
5085:
5082:
5080:
5077:
5075:
5072:
5070:
5067:
5066:
5061:
5058:
5054:
5050:
5048:
5045:
5041:
5037:
5033:
5029:
5025:
5021:
5020:
5015:
5012:
5008:
5004:
5002:
4999:
4995:
4991:
4987:
4983:
4979:
4975:
4974:
4972:
4964:
4961:
4958:
4955:
4951:
4946:
4941:
4938:
4933:
4917:
4916:
4915:
4914:
4901:
4898:
4893:
4886:
4883:
4880:
4876:
4872:
4869:
4866:
4863:
4858:
4855:
4850:
4829:
4814:
4811:
4807:
4806:
4793:
4788:
4783:
4778:
4773:
4768:
4765:
4760:
4755:
4750:
4745:
4740:
4735:
4732:
4727:
4722:
4717:
4712:
4707:
4702:
4699:
4685:
4684:
4673:
4668:
4663:
4658:
4653:
4648:
4643:
4640:
4635:
4630:
4625:
4620:
4615:
4595:
4594:
4583:
4580:
4577:
4572:
4567:
4562:
4557:
4552:
4528:
4527:
4516:
4511:
4506:
4501:
4496:
4491:
4486:
4481:
4476:
4471:
4468:
4465:
4462:
4457:
4452:
4447:
4444:
4439:
4434:
4429:
4426:
4421:
4416:
4411:
4397:
4396:
4383:
4377:
4374:
4372:
4369:
4368:
4365:
4362:
4357:
4355:
4352:
4351:
4348:
4345:
4343:
4340:
4339:
4337:
4302:exterior power
4281:
4278:
4262:
4261:
4247:
4244:
4241:
4237:
4232:
4228:
4223:
4220:
4217:
4213:
4208:
4204:
4199:
4195:
4191:
4186:
4183:
4180:
4176:
4171:
4168:
4164:
4090:
4089:
4070:
4063:
4032:
4018:
3974:
3961:elements, and
3896:
3893:
3868:
3864:
3858:
3853:
3850:
3847:
3843:
3839:
3834:
3830:
3824:
3819:
3816:
3813:
3809:
3804:
3800:
3797:
3794:
3783:
3782:
3771:
3768:
3763:
3759:
3755:
3752:
3749:
3744:
3740:
3736:
3730:
3727:
3723:
3719:
3716:
3711:
3706:
3702:
3699:
3696:
3693:
3687:
3683:
3680:
3677:
3673:
3668:
3663:
3659:
3656:
3653:
3650:
3647:
3641:
3637:
3633:
3628:
3623:
3619:
3616:
3613:
3606:
3602:
3597:
3593:
3590:
3587:
3584:
3581:
3574:
3570:
3565:
3561:
3558:
3555:
3552:
3547:
3543:
3539:
3536:
3533:
3528:
3524:
3520:
3516:
3512:
3507:
3504:
3501:
3497:
3491:
3488:
3483:
3478:
3449:
3445:
3441:
3438:
3435:
3430:
3426:
3414:
3413:
3402:
3396:
3392:
3389:
3386:
3382:
3377:
3373:
3370:
3367:
3361:
3357:
3353:
3348:
3344:
3341:
3334:
3330:
3325:
3319:
3316:
3311:
3306:
3303:
3300:
3297:
3294:
3287:
3283:
3278:
3272:
3269:
3264:
3259:
3256:
3253:
3250:
3245:
3241:
3237:
3234:
3231:
3226:
3222:
3218:
3213:
3210:
3207:
3203:
3197:
3194:
3189:
3184:
3160:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3126:
3118:
3114:
3110:
3105:
3101:
3096:
3092:
3088:
3084:
3081:
3076:
3073:
3070:
3066:
3061:
3057:
3023:
3020:
3017:
3013:
3008:
3004:
2957:
2956:
2945:
2938:
2934:
2926:
2923:
2919:
2915:
2912:
2907:
2902:
2898:
2892:
2889:
2884:
2881:
2878:
2874:
2868:
2865:
2860:
2855:
2843:matrix). Then
2820:
2817:
2812:
2808:
2804:
2801:
2798:
2793:
2789:
2785:
2780:
2776:
2772:
2769:
2749:
2746:
2741:
2737:
2733:
2730:
2727:
2722:
2718:
2714:
2709:
2705:
2701:
2698:
2672:
2671:
2660:
2654:
2648:
2640:
2636:
2632:
2629:
2626:
2622:
2617:
2612:
2609:
2604:
2580:
2579:
2566:
2558:
2555:
2551:
2547:
2545:
2542:
2538:
2535:
2531:
2527:
2523:
2520:
2516:
2512:
2511:
2508:
2505:
2503:
2500:
2498:
2495:
2493:
2490:
2489:
2484:
2481:
2477:
2473:
2471:
2468:
2464:
2460:
2456:
2452:
2448:
2444:
2443:
2438:
2435:
2431:
2427:
2425:
2422:
2418:
2414:
2410:
2406:
2402:
2398:
2397:
2395:
2390:
2386:
2361:is called the
2343:Main article:
2340:
2337:
2336:
2335:
2322:
2319:
2315:
2309:
2306:
2303:
2299:
2295:
2292:
2289:
2284:
2281:
2277:
2271:
2266:
2263:
2260:
2256:
2252:
2247:
2244:
2240:
2234:
2231:
2227:
2221:
2216:
2213:
2210:
2206:
2202:
2197:
2194:
2190:
2184:
2181:
2177:
2173:
2170:
2167:
2162:
2159:
2155:
2149:
2146:
2142:
2138:
2133:
2130:
2126:
2120:
2117:
2113:
2109:
2104:
2101:
2097:
2091:
2088:
2084:
2080:
2077:
2073:
2069:
2066:
2045:
2044:
2031:
2028:
2024:
2018:
2015:
2012:
2008:
2004:
2001:
1998:
1993:
1990:
1986:
1980:
1975:
1972:
1969:
1965:
1961:
1956:
1953:
1949:
1943:
1940:
1936:
1930:
1925:
1922:
1919:
1915:
1911:
1906:
1903:
1899:
1893:
1890:
1886:
1882:
1879:
1876:
1871:
1868:
1864:
1858:
1855:
1851:
1847:
1842:
1839:
1835:
1829:
1826:
1822:
1818:
1813:
1810:
1806:
1800:
1797:
1793:
1789:
1786:
1782:
1778:
1775:
1743:
1740:
1736:
1730:
1727:
1724:
1720:
1716:
1713:
1710:
1707:
1702:
1699:
1695:
1681:, denoted det(
1666:
1661:
1658:
1654:
1650:
1647:
1644:
1615:Main article:
1612:
1609:
1607:
1604:
1598:
1591:
1568:
1565:, of a minor,
1561:
1555:
1552:
1538:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1510:
1505:
1502:
1499:
1495:
1491:
1486:
1482:
1479:
1474:
1471:
1468:
1464:
1391:
1388:
1385:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1341:
1316:
1312:
1308:
1305:
1302:
1297:
1293:
1289:
1284:
1280:
1276:
1271:
1267:
1263:
1260:
1257:
1252:
1248:
1244:
1239:
1235:
1230:
1207:
1204:
1201:
1197:
1174:
1171:
1168:
1164:
1160:
1157:
1135:
1132:
1129:
1125:
1121:
1101:
1096:
1093:
1090:
1086:
1064:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1030:
1022:
1018:
1014:
1009:
1005:
1000:
996:
992:
988:
960:
957:
952:
948:
944:
941:
938:
933:
929:
925:
920:
916:
912:
909:
889:
886:
881:
877:
873:
870:
867:
862:
858:
854:
849:
845:
841:
838:
796:
791:
788:
783:
777:
771:
766:
763:
758:
646:, also called
592:
589:
588:
587:
576:
573:
570:
567:
562:
559:
556:
552:
548:
543:
540:
537:
533:
529:
526:
523:
520:
515:
512:
509:
505:
487:
486:
475:
472:
469:
466:
463:
460:
457:
454:
451:
446:
440:
437:
435:
432:
429:
428:
425:
422:
420:
417:
416:
414:
409:
406:
401:
395:
392:
390:
387:
385:
382:
379:
378:
375:
372:
370:
367:
365:
362:
361:
358:
355:
353:
350:
348:
345:
344:
342:
337:
334:
329:
326:
323:
319:
303:
296:
290:
289:
276:
270:
267:
265:
262:
260:
257:
254:
253:
250:
247:
245:
242:
240:
237:
236:
233:
230:
228:
225:
223:
220:
219:
217:
187:
184:
181:
177:
173:
170:
167:
141:
86:
83:
81:
78:
39:linear algebra
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5842:
5831:
5828:
5826:
5825:Matrix theory
5823:
5822:
5820:
5805:
5797:
5796:
5793:
5787:
5784:
5782:
5781:Sparse matrix
5779:
5777:
5774:
5772:
5769:
5767:
5764:
5763:
5761:
5759:
5755:
5749:
5746:
5744:
5741:
5739:
5736:
5734:
5731:
5729:
5726:
5724:
5721:
5720:
5718:
5716:constructions
5715:
5711:
5705:
5704:Outermorphism
5702:
5700:
5697:
5695:
5692:
5690:
5687:
5685:
5682:
5680:
5677:
5675:
5672:
5670:
5667:
5665:
5664:Cross product
5662:
5660:
5657:
5656:
5654:
5652:
5648:
5642:
5639:
5637:
5634:
5632:
5631:Outer product
5629:
5627:
5624:
5622:
5619:
5617:
5614:
5612:
5611:Orthogonality
5609:
5608:
5606:
5604:
5600:
5594:
5591:
5589:
5588:Cramer's rule
5586:
5584:
5581:
5579:
5576:
5574:
5571:
5569:
5566:
5564:
5561:
5559:
5558:Decomposition
5556:
5554:
5551:
5550:
5548:
5546:
5542:
5537:
5527:
5524:
5522:
5519:
5517:
5514:
5512:
5509:
5507:
5504:
5502:
5499:
5497:
5494:
5492:
5489:
5487:
5484:
5482:
5479:
5477:
5474:
5472:
5469:
5467:
5464:
5462:
5459:
5457:
5454:
5452:
5449:
5447:
5444:
5442:
5439:
5437:
5434:
5433:
5431:
5427:
5421:
5418:
5416:
5413:
5412:
5409:
5405:
5398:
5393:
5391:
5386:
5384:
5379:
5378:
5375:
5369:
5368:
5363:
5361:
5360:
5355:
5352:
5349:
5348:
5344:
5336:
5332:
5327:
5324:
5319:
5313:
5309:
5308:
5300:
5297:
5293:
5292:0-521-66402-0
5289:
5285:
5284:
5277:
5274:
5271:
5267:
5261:
5258:
5253:
5252:
5244:
5242:
5240:
5236:
5233:
5229:
5223:
5221:
5217:
5213:
5212:
5205:
5202:
5195:
5191:
5188:
5186:
5183:
5182:
5178:
5176:
5174:
5170:
5166:
5162:
5141:
5133:
5130:
5126:
5120:
5113:
5110:
5106:
5098:
5095:
5091:
5083:
5078:
5073:
5068:
5059:
5056:
5052:
5046:
5039:
5035:
5027:
5023:
5013:
5010:
5006:
5000:
4993:
4989:
4981:
4977:
4970:
4959:
4949:
4944:
4939:
4936:
4922:
4921:
4920:
4899:
4896:
4884:
4881:
4878:
4870:
4867:
4861:
4856:
4853:
4839:
4838:
4837:
4836:
4835:
4832:
4828:
4824:
4820:
4812:
4810:
4791:
4781:
4776:
4766:
4763:
4758:
4748:
4743:
4733:
4730:
4725:
4715:
4710:
4700:
4697:
4690:
4689:
4688:
4671:
4666:
4656:
4651:
4641:
4638:
4633:
4623:
4618:
4604:
4603:
4602:
4600:
4599:antisymmetric
4581:
4578:
4575:
4570:
4560:
4555:
4541:
4540:
4539:
4537:
4533:
4509:
4499:
4494:
4484:
4479:
4469:
4463:
4455:
4445:
4442:
4437:
4427:
4424:
4419:
4402:
4401:
4400:
4381:
4375:
4370:
4363:
4360:
4353:
4346:
4341:
4335:
4326:
4325:
4324:
4322:
4318:
4314:
4310:
4305:
4303:
4299:
4295:
4291:
4290:wedge product
4287:
4279:
4277:
4275:
4271:
4267:
4245:
4242:
4239:
4221:
4218:
4215:
4197:
4193:
4189:
4184:
4181:
4178:
4155:
4154:
4153:
4151:
4147:
4143:
4140:elements and
4139:
4135:
4131:
4127:
4123:
4119:
4115:
4111:
4107:
4103:
4099:
4095:
4087:
4083:
4079:
4075:
4071:
4068:
4064:
4061:
4057:
4053:
4049:
4045:
4041:
4037:
4033:
4030:
4025:
4021:
4016:
4012:
4008:
4007:
4006:
4004:
4000:
3996:
3991:
3987:
3981:
3977:
3972:
3968:
3964:
3960:
3956:
3952:
3948:
3944:
3940:
3936:
3931:
3929:
3925:
3921:
3918:
3914:
3910:
3906:
3902:
3894:
3892:
3890:
3886:
3866:
3862:
3856:
3851:
3848:
3845:
3841:
3837:
3832:
3828:
3822:
3817:
3814:
3811:
3807:
3798:
3795:
3769:
3761:
3757:
3753:
3750:
3747:
3742:
3738:
3728:
3725:
3721:
3717:
3714:
3697:
3694:
3685:
3681:
3678:
3675:
3671:
3666:
3654:
3651:
3648:
3639:
3635:
3631:
3626:
3614:
3604:
3600:
3595:
3588:
3585:
3582:
3572:
3568:
3563:
3556:
3553:
3545:
3541:
3537:
3534:
3531:
3526:
3522:
3505:
3502:
3499:
3489:
3486:
3469:
3468:
3467:
3465:
3447:
3443:
3439:
3436:
3433:
3428:
3424:
3400:
3394:
3390:
3387:
3384:
3380:
3375:
3371:
3368:
3365:
3359:
3355:
3351:
3346:
3342:
3332:
3328:
3323:
3317:
3314:
3301:
3298:
3295:
3285:
3281:
3276:
3270:
3267:
3254:
3251:
3243:
3239:
3235:
3232:
3229:
3224:
3220:
3211:
3208:
3205:
3195:
3192:
3175:
3174:
3173:
3158:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3116:
3112:
3108:
3103:
3099:
3094:
3086:
3079:
3074:
3071:
3068:
3047:
3043:
3039:
3021:
3018:
3015:
2994:
2990:
2986:
2982:
2978:
2974:
2970:
2966:
2962:
2943:
2924:
2921:
2917:
2913:
2910:
2890:
2887:
2882:
2879:
2876:
2866:
2863:
2846:
2845:
2844:
2842:
2838:
2834:
2818:
2815:
2810:
2806:
2802:
2799:
2796:
2791:
2787:
2783:
2778:
2774:
2770:
2767:
2747:
2744:
2739:
2735:
2731:
2728:
2725:
2720:
2716:
2712:
2707:
2703:
2699:
2696:
2687:
2685:
2681:
2677:
2658:
2627:
2624:
2620:
2615:
2610:
2607:
2593:
2592:
2591:
2589:
2585:
2564:
2556:
2553:
2549:
2543:
2536:
2533:
2529:
2521:
2518:
2514:
2506:
2501:
2496:
2491:
2482:
2479:
2475:
2469:
2462:
2458:
2450:
2446:
2436:
2433:
2429:
2423:
2416:
2412:
2404:
2400:
2393:
2388:
2376:
2375:
2374:
2372:
2368:
2364:
2360:
2356:
2355:Cramer's rule
2352:
2346:
2338:
2320:
2317:
2313:
2307:
2304:
2301:
2293:
2290:
2282:
2279:
2275:
2269:
2264:
2261:
2258:
2254:
2250:
2245:
2242:
2238:
2232:
2229:
2225:
2219:
2214:
2211:
2208:
2204:
2200:
2195:
2192:
2188:
2182:
2179:
2175:
2171:
2168:
2165:
2160:
2157:
2153:
2147:
2144:
2140:
2136:
2131:
2128:
2124:
2118:
2115:
2111:
2107:
2102:
2099:
2095:
2089:
2086:
2082:
2078:
2054:
2053:
2052:
2050:
2029:
2026:
2022:
2016:
2013:
2010:
2002:
1999:
1991:
1988:
1984:
1978:
1973:
1970:
1967:
1963:
1959:
1954:
1951:
1947:
1941:
1938:
1934:
1928:
1923:
1920:
1917:
1913:
1909:
1904:
1901:
1897:
1891:
1888:
1884:
1880:
1877:
1874:
1869:
1866:
1862:
1856:
1853:
1849:
1845:
1840:
1837:
1833:
1827:
1824:
1820:
1816:
1811:
1808:
1804:
1798:
1795:
1791:
1787:
1763:
1762:
1761:
1759:
1741:
1738:
1734:
1728:
1725:
1722:
1714:
1711:
1705:
1700:
1697:
1693:
1684:
1680:
1659:
1656:
1652:
1645:
1642:
1633:
1629:
1624:
1618:
1610:
1605:
1603:
1601:
1594:
1592:ijk...,pqr...
1587:
1583:
1579:
1575:
1571:
1569:ijk...,pqr...
1564:
1562:ijk...,pqr...
1553:
1551:
1536:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1508:
1503:
1500:
1497:
1493:
1489:
1484:
1477:
1472:
1469:
1466:
1462:
1453:
1449:
1445:
1441:
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1409:
1405:
1386:
1358:
1352:
1346:
1339:
1314:
1310:
1306:
1303:
1300:
1295:
1291:
1287:
1282:
1278:
1274:
1269:
1265:
1261:
1258:
1255:
1250:
1246:
1242:
1237:
1233:
1228:
1205:
1202:
1199:
1195:
1172:
1169:
1166:
1158:
1133:
1130:
1127:
1123:
1099:
1094:
1091:
1088:
1062:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1020:
1016:
1012:
1007:
1003:
998:
990:
978:
974:
958:
955:
950:
946:
942:
939:
936:
931:
927:
923:
918:
914:
910:
907:
887:
884:
879:
875:
871:
868:
865:
860:
856:
852:
847:
843:
839:
836:
827:
825:
821:
817:
813:
789:
786:
775:
764:
761:
745:
741:
737:
733:
729:
725:
721:
717:
713:
709:
705:
701:
697:
693:
689:
685:
681:
677:
673:
669:
665:
661:
657:
653:
649:
645:
641:
638:
634:
630:
626:
622:
618:
614:
610:
606:
602:
598:
590:
574:
571:
568:
560:
557:
554:
550:
541:
538:
535:
527:
524:
518:
513:
510:
507:
503:
492:
491:
490:
473:
470:
464:
461:
455:
452:
449:
444:
438:
433:
430:
423:
418:
412:
404:
399:
393:
388:
383:
380:
373:
368:
363:
356:
351:
346:
340:
332:
327:
324:
321:
317:
309:
308:
307:
302:
295:
274:
268:
263:
258:
255:
248:
243:
238:
231:
226:
221:
215:
206:
205:
204:
201:
185:
182:
179:
171:
168:
157:
153:
149:
144:
140:
136:
132:
128:
124:
120:
116:
112:
108:
104:
100:
96:
92:
84:
79:
77:
75:
71:
67:
63:
59:
58:square matrix
55:
51:
48:
44:
40:
33:
19:
5830:Determinants
5714:Vector space
5567:
5446:Vector space
5366:
5358:
5334:
5326:
5306:
5299:
5282:
5276:
5260:
5250:
5209:
5204:
5160:
5158:
4918:
4830:
4826:
4822:
4818:
4816:
4808:
4686:
4596:
4529:
4398:
4320:
4316:
4312:
4308:
4306:
4297:
4293:
4288:, using the
4283:
4273:
4269:
4265:
4263:
4149:
4145:
4141:
4137:
4133:
4125:
4121:
4117:
4113:
4109:
4105:
4101:
4091:
4066:
4059:
4055:
4051:
4047:
4043:
4039:
4028:
4027:is called a
4023:
4019:
4014:
4010:
4002:
3998:
3994:
3989:
3985:
3979:
3975:
3970:
3966:
3962:
3958:
3954:
3946:
3942:
3938:
3934:
3932:
3927:
3923:
3919:
3907:matrix with
3904:
3900:
3898:
3888:
3884:
3784:
3463:
3415:
3045:
3041:
3037:
2992:
2988:
2984:
2980:
2976:
2972:
2968:
2964:
2960:
2958:
2840:
2836:
2832:
2688:
2683:
2679:
2673:
2587:
2583:
2581:
2370:
2366:
2362:
2358:
2348:
2048:
2046:
1757:
1682:
1678:
1631:
1627:
1620:
1596:
1589:
1585:
1581:
1577:
1573:
1566:
1559:
1557:
1451:
1447:
1443:
1439:
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1407:
1403:
976:
972:
828:
824:zeroth minor
823:
819:
815:
811:
743:
739:
735:
731:
727:
723:
719:
715:
711:
707:
703:
699:
695:
691:
690:by deleting
687:
683:
679:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
636:
632:
628:
624:
620:
616:
615:with 0 <
608:
604:
600:
596:
594:
488:
300:
293:
291:
202:
155:
151:
147:
142:
138:
134:
130:
118:
114:
110:
106:
102:
98:
94:
90:
88:
85:First minors
69:
66:first minors
65:
61:
49:
42:
36:
5694:Multivector
5659:Determinant
5616:Dot product
5461:Linear span
4536:alternating
4067:basic minor
1376:(where the
607:matrix and
123:determinant
119:first minor
54:determinant
32:Graph minor
5819:Categories
5728:Direct sum
5563:Invertible
5466:Linear map
5196:References
4268:of {1,...,
4132:of {1,...,
3953:of {1,...,
1554:Complement
5758:Numerical
5521:Transpose
5359:Cofactors
5248:"Minor".
5185:Submatrix
5121:⋯
5084:⋮
5079:⋱
5074:⋮
5069:⋮
5047:⋯
5001:⋯
4937:−
4868:−
4821:the term
4782:∧
4749:∧
4731:−
4716:∧
4698:−
4657:∧
4642:−
4624:∧
4561:∧
4485:−
4464:∧
4361:−
4194:∑
4050:. For an
4036:submatrix
3993:minor of
3899:Given an
3842:∑
3838:−
3808:∑
3796:−
3754:∧
3751:…
3748:∧
3698:±
3679:−
3655:∧
3652:…
3649:∧
3615:∧
3589:∧
3586:…
3583:∧
3557:±
3538:∧
3535:…
3532:∧
3487:−
3437:…
3388:−
3372:∧
3369:…
3366:∧
3343:∧
3315:−
3302:∧
3299:…
3296:∧
3268:−
3255:±
3236:∧
3233:…
3230:∧
3193:−
3147:…
2891:±
2864:−
2816:≤
2800:…
2771:≤
2745:≤
2729:…
2700:≤
2628:
2608:−
2544:⋯
2507:⋮
2502:⋱
2497:⋮
2492:⋮
2470:⋯
2424:⋯
2291:−
2255:∑
2205:∑
2169:⋯
2000:−
1964:∑
1914:∑
1878:⋯
1712:−
1529:≠
1517:≠
1304:…
1259:…
1051:…
956:≤
940:⋯
911:≤
885:≤
869:⋯
840:≤
776:⋅
726:rows and
698:rows and
572:−
525:−
462:−
456:−
431:−
394:◻
381:−
374:◻
369:◻
364:◻
357:◻
256:−
169:−
127:submatrix
121:) is the
70:cofactors
5804:Category
5743:Subspace
5738:Quotient
5689:Bivector
5603:Bilinear
5545:Matrices
5420:Glossary
5179:See also
5165:adjugate
4819:cofactor
4532:bilinear
4124:matrix,
4112:matrix,
4096:and the
3983:for the
3945:matrix,
3729:′
3718:′
3686:′
3640:′
3395:′
3360:′
3048:. Also,
2925:′
2914:′
2676:adjugate
2371:comatrix
156:cofactor
5415:Outline
5169:adjoint
5163:is not
5161:adjunct
4823:adjunct
4272:} with
4148:} with
4136:} with
4017:, then
3969:} with
3957:} with
1635:matrix
654:or, if
613:integer
146:. The (
125:of the
117:, or a
74:inverse
52:is the
5699:Tensor
5511:Kernel
5441:Vector
5436:Scalar
5314:
5290:
5268:
5230:
4292:: the
4130:subset
4116:is an
4104:is an
4084:. See
3951:subset
3937:is an
3915:) and
3416:where
2995:) and
2959:where
2835:is an
2062:
1771:
1586:pqr...
1582:ijk...
818:. The
623:, and
599:be an
500:
47:matrix
5568:Minor
5553:Block
5491:Basis
5367:Minor
4304:map.
4128:is a
3949:is a
3913:field
2682:) of
640:minor
115:minor
95:minor
45:of a
43:minor
5723:Dual
5578:Rank
5312:ISBN
5288:ISBN
5266:ISBN
5228:ISBN
4597:and
4534:and
4072:For
3917:rank
3909:real
3887:and
2991:and
2803:<
2797:<
2784:<
2760:and
2732:<
2726:<
2713:<
1426:and
1410:and
975:and
943:<
937:<
924:<
900:and
872:<
866:<
853:<
829:Let
631:. A
595:Let
41:, a
5167:or
4954:det
4300:th
4046:or
4009:If
3511:det
3083:det
2983:or
2933:det
2625:det
2373:):
2065:det
1774:det
1481:det
1331:or
1220:or
1187:or
1148:or
1120:det
1112:or
1085:det
987:det
738:of
674:of
662:, (
650:of
642:of
611:an
575:13.
408:det
336:det
304:2,3
297:2,3
143:i,j
89:If
37:In
5821::
5333:,
5238:^
5219:^
5175:.
5040:22
5028:12
4994:21
4982:11
4831:ij
4701:13
4601:,
4538:,
4065:A
4013:=
4005:.
3891:.
2993:J′
2985:I′
2971:,
2965:J′
2963:,
2961:I′
2686:.
2590::
2463:22
2451:21
2417:12
2405:11
1599:ij
1450:,
658:=
627:≤
619:≤
474:13
269:11
200:.
154:)
150:,
113:)
109:,
5396:e
5389:t
5382:v
5320:.
5294:.
5254:.
5214:.
5142:]
5134:n
5131:n
5127:A
5114:n
5111:2
5107:A
5099:n
5096:1
5092:A
5060:2
5057:n
5053:A
5036:A
5024:A
5014:1
5011:n
5007:A
4990:A
4978:A
4971:[
4963:)
4960:M
4957:(
4950:1
4945:=
4940:1
4932:M
4900:j
4897:i
4892:M
4885:j
4882:+
4879:i
4875:)
4871:1
4865:(
4862:=
4857:j
4854:i
4849:A
4827:A
4792:3
4787:e
4777:2
4772:e
4767:5
4764:+
4759:3
4754:e
4744:1
4739:e
4734:7
4726:2
4721:e
4711:1
4706:e
4672:,
4667:i
4662:e
4652:j
4647:e
4639:=
4634:j
4629:e
4619:i
4614:e
4582:,
4579:0
4576:=
4571:i
4566:e
4556:i
4551:e
4515:)
4510:3
4505:e
4500:+
4495:2
4490:e
4480:1
4475:e
4470:4
4467:(
4461:)
4456:3
4451:e
4446:2
4443:+
4438:2
4433:e
4428:3
4425:+
4420:1
4415:e
4410:(
4382:)
4376:1
4371:2
4364:1
4354:3
4347:4
4342:1
4336:(
4321:k
4317:k
4313:k
4309:k
4298:k
4294:k
4274:k
4270:n
4266:K
4246:J
4243:,
4240:K
4236:]
4231:B
4227:[
4222:K
4219:,
4216:I
4212:]
4207:A
4203:[
4198:K
4190:=
4185:J
4182:,
4179:I
4175:]
4170:B
4167:A
4163:[
4150:k
4146:p
4142:J
4138:k
4134:m
4126:I
4122:p
4118:n
4114:B
4110:n
4106:m
4102:A
4060:n
4056:n
4052:n
4040:k
4031:.
4024:J
4022:,
4020:I
4015:J
4011:I
4003:J
3999:I
3995:A
3990:k
3986:k
3980:J
3978:,
3976:I
3971:k
3967:n
3963:J
3959:k
3955:m
3947:I
3943:n
3939:m
3935:A
3928:r
3924:r
3920:r
3905:n
3901:m
3889:J
3885:I
3867:s
3863:j
3857:k
3852:1
3849:=
3846:s
3833:s
3829:i
3823:k
3818:1
3815:=
3812:s
3803:)
3799:1
3793:(
3770:.
3767:)
3762:n
3758:e
3743:1
3739:e
3735:(
3726:I
3722:,
3715:J
3710:]
3705:A
3701:[
3695:=
3692:)
3682:k
3676:n
3672:i
3667:e
3662:A
3658:(
3646:)
3636:1
3632:i
3627:e
3622:A
3618:(
3612:)
3605:k
3601:j
3596:e
3592:(
3580:)
3573:1
3569:j
3564:e
3560:(
3554:=
3551:)
3546:n
3542:e
3527:1
3523:e
3519:(
3515:A
3506:J
3503:,
3500:I
3496:]
3490:1
3482:A
3477:[
3464:A
3448:n
3444:e
3440:,
3434:,
3429:1
3425:e
3401:,
3391:k
3385:n
3381:i
3376:e
3356:1
3352:i
3347:e
3340:)
3333:k
3329:j
3324:e
3318:1
3310:A
3305:(
3293:)
3286:1
3282:j
3277:e
3271:1
3263:A
3258:(
3252:=
3249:)
3244:n
3240:e
3225:1
3221:e
3217:(
3212:J
3209:,
3206:I
3202:]
3196:1
3188:A
3183:[
3159:)
3153:k
3150:,
3144:,
3141:1
3138:=
3135:q
3132:,
3129:p
3125:)
3117:q
3113:j
3109:,
3104:p
3100:i
3095:A
3091:(
3087:(
3080:=
3075:J
3072:,
3069:I
3065:]
3060:A
3056:[
3046:J
3042:I
3038:A
3022:J
3019:,
3016:I
3012:]
3007:A
3003:[
2989:J
2981:I
2977:n
2973:J
2969:I
2944:,
2937:A
2922:I
2918:,
2911:J
2906:]
2901:A
2897:[
2888:=
2883:J
2880:,
2877:I
2873:]
2867:1
2859:A
2854:[
2841:n
2837:n
2833:A
2819:n
2811:k
2807:j
2792:2
2788:j
2779:1
2775:j
2768:1
2748:n
2740:k
2736:i
2721:2
2717:i
2708:1
2704:i
2697:1
2684:A
2659:.
2653:T
2647:C
2639:)
2635:A
2631:(
2621:1
2616:=
2611:1
2603:A
2588:A
2584:A
2565:]
2557:n
2554:n
2550:C
2537:2
2534:n
2530:C
2522:1
2519:n
2515:C
2483:n
2480:2
2476:C
2459:C
2447:C
2437:n
2434:1
2430:C
2413:C
2401:C
2394:[
2389:=
2385:C
2359:A
2321:j
2318:i
2314:M
2308:j
2305:+
2302:i
2298:)
2294:1
2288:(
2283:j
2280:i
2276:a
2270:n
2265:1
2262:=
2259:j
2251:=
2246:j
2243:i
2239:C
2233:j
2230:i
2226:a
2220:n
2215:1
2212:=
2209:j
2201:=
2196:n
2193:i
2189:C
2183:n
2180:i
2176:a
2172:+
2166:+
2161:3
2158:i
2154:C
2148:3
2145:i
2141:a
2137:+
2132:2
2129:i
2125:C
2119:2
2116:i
2112:a
2108:+
2103:1
2100:i
2096:C
2090:1
2087:i
2083:a
2079:=
2076:)
2072:A
2068:(
2049:i
2030:j
2027:i
2023:M
2017:j
2014:+
2011:i
2007:)
2003:1
1997:(
1992:j
1989:i
1985:a
1979:n
1974:1
1971:=
1968:i
1960:=
1955:j
1952:i
1948:C
1942:j
1939:i
1935:a
1929:n
1924:1
1921:=
1918:i
1910:=
1905:j
1902:n
1898:C
1892:j
1889:n
1885:a
1881:+
1875:+
1870:j
1867:3
1863:C
1857:j
1854:3
1850:a
1846:+
1841:j
1838:2
1834:C
1828:j
1825:2
1821:a
1817:+
1812:j
1809:1
1805:C
1799:j
1796:1
1792:a
1788:=
1785:)
1781:A
1777:(
1758:j
1742:j
1739:i
1735:M
1729:j
1726:+
1723:i
1719:)
1715:1
1709:(
1706:=
1701:j
1698:i
1694:C
1683:A
1679:A
1665:)
1660:j
1657:i
1653:a
1649:(
1646:=
1643:A
1632:n
1628:n
1597:a
1590:M
1578:A
1574:A
1567:M
1560:B
1537:)
1532:j
1526:q
1523:,
1520:i
1514:p
1509:)
1504:q
1501:,
1498:p
1494:A
1490:(
1485:(
1478:=
1473:j
1470:,
1467:i
1463:M
1452:j
1448:i
1444:J
1440:I
1436:J
1432:I
1428:J
1424:I
1420:J
1416:I
1412:J
1408:I
1404:I
1390:)
1387:i
1384:(
1362:)
1359:j
1356:(
1353:,
1350:)
1347:i
1344:(
1340:M
1315:k
1311:j
1307:,
1301:,
1296:2
1292:j
1288:,
1283:1
1279:j
1275:,
1270:k
1266:i
1262:,
1256:,
1251:2
1247:i
1243:,
1238:1
1234:i
1229:M
1206:J
1203:,
1200:I
1196:M
1173:J
1170:,
1167:I
1163:]
1159:A
1156:[
1134:J
1131:,
1128:I
1124:A
1100:A
1095:J
1092:,
1089:I
1063:)
1057:k
1054:,
1048:,
1045:1
1042:=
1039:q
1036:,
1033:p
1029:)
1021:q
1017:j
1013:,
1008:p
1004:i
999:A
995:(
991:(
977:J
973:I
959:n
951:k
947:j
932:2
928:j
919:1
915:j
908:1
888:m
880:k
876:i
861:2
857:i
848:1
844:i
837:1
816:k
812:k
795:)
790:k
787:n
782:(
770:)
765:k
762:m
757:(
744:A
740:A
732:k
730:−
728:n
724:k
722:−
720:m
716:A
712:k
708:k
704:k
702:−
700:n
696:k
694:−
692:m
688:A
684:k
680:k
676:A
670:)
668:k
666:−
664:n
660:n
656:m
652:A
644:A
637:k
633:k
629:n
625:k
621:m
617:k
609:k
605:n
601:m
597:A
569:=
566:)
561:3
558:,
555:2
551:M
547:(
542:3
539:+
536:2
532:)
528:1
522:(
519:=
514:3
511:,
508:2
504:C
471:=
468:)
465:4
459:(
453:9
450:=
445:]
439:9
434:1
424:4
419:1
413:[
405:=
400:]
389:9
384:1
352:4
347:1
341:[
333:=
328:3
325:,
322:2
318:M
301:C
294:M
275:]
264:9
259:1
249:5
244:0
239:3
232:7
227:4
222:1
216:[
186:j
183:+
180:i
176:)
172:1
166:(
152:j
148:i
139:M
135:j
131:i
111:j
107:i
103:j
99:i
91:A
62:A
50:A
34:.
20:)
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