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**Extra info for An elementary treatise on cubic and quartic curves**

**Sample text**

5 The Adjoint Determinant 37 Proof. 5 on the product of two determinants, n bij = air Ajr r=1 = δij A. Hence, |bij |n = diag|A A . . A|n = An . The theorem follows immediately if A = 0. 16) with a change in notation, |Aij |n = 0, that is, adj A = 0. Hence, the Cauchy identity is valid for all A. 3 An Identity Involving a Hybrid Determinant Let An = |aij |n and Bn = |bij |n , and let Hij denote the hybrid determinant formed by replacing the jth row of An by the ith row of Bn . Then, n Hij = bis Ajs .

The cofactor is zero when any row or column parameter is less than 1 or greater than n. Illustration. (4) (4) (4) (6) (6) (4) (4) A12,23 = −A21,23 = −A12,32 = A21,32 = M12,23 = N34,14 , (6) (6) (6) A135,235 = −A135,253 = A135,523 = A315,253 = −M135,235 = −N246,146 , (n) (n) (n) Ai2 i1 i3 ;j1 j2 j3 = −Ai1 i2 i3 ;j1 j2 j3 = Ai1 i2 i3 ;j1 j3 j2 , (n) Ai1 i2 i3 ;j1 j2 (n−p) = 0 if p < 0 or p ≥ n or p = n − j1 or p = n − j2 . 3 The Expansion of Cofactors in Terms of Higher Cofactors (n) Since the ﬁrst cofactor Aip is itself a determinant of order (n − 1), it can be expanded by the (n − 1) elements from any row or column and their ﬁrst (n) cofactors.

R rows .. ar,r+1 A . r = . . . . . . . . . . . . . . . . . . . . ar+1,r+1 . . ar+1,n (n − r) rows .. .. . .. a . a n,r+1 = Ar ar+1,r+1 .. an,r+1 r nn . . ar+1,n .. ... r . The ﬁrst stage of the proof follows. The second stage proceeds as follows. 1) appear 40 3. Intermediate Determinant Theory as a block in the top left-hand corner. Denote the result by (adj A)∗ . Then, (adj A)∗ = σ adj A, where σ = (−1)(p1 −1)+(p2 −2)+···+(pr −r)+(q1 −1)+(q2 −2)+···+(qr −r) = (−1)(p1 +p2 +···+pr )+(q1 +q2 +···+qr ) .