By Vein R., Dale P.

ISBN-10: 0387985581

ISBN-13: 9780387985589

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Ir are omitted. 26 3. jr e1 · · · en , where, in this case, the symbol ∗ denotes that those vectors with suﬃxes j1 , j2 , . . , jr are omitted. jr ej1 · · · ejr = ∗ e1 · · · en . 2), it is found that ej1 · · · ejr ∗ e1 · · · en = (−1)q (e1 · · · en ), where n q= s=1 js − 12 r(r + 1). jr e1 · · · en . ir which is the general form of the Laplace expansion of An in which the sum extends over the row parameters. By a similar argument, it can be shown that An is also equal to the same expression in which the sum extends over the column parameters.

R=1 s=1 Note that (2) and (3) can be obtained formally from (B) and (D), respectively, by interchanging the symbols a and A and either raising or lowering all their parameters. 1 The Adjoint Determinant Deﬁnition The adjoint of a matrix A = [aij ]n is denoted by adj A and is deﬁned by adj A = [Aji ]n . The adjoint or adjugate or a determinant A = |aij |n = det A is denoted by adj A and is deﬁned by adj A = |Aji |n = |Aij |n = det(adj A). 2 The Cauchy Identity The following theorem due to Cauchy is valid for all determinants.

1 b11 b21 ... bn1 b12 b22 ... bn2 . . b1n . . b2n ... . . bnn . 15) 2n Reduce all the elements in the ﬁrst n rows and the ﬁrst n columns, at present occupied by the aij , to zero by means of the row operations n Ri = Ri + aij Rn+j , j=1 1 ≤ i ≤ n. 16) 34 3. Intermediate Determinant Theory The result is: c11 c21 ... cn1 An Bn = −1 b11 −1 b21 ... −1 bn1 c12 c22 ... cn2 b12 b22 ... bn2 . . c1n . . c2n ... . . cnn . . b1n . . b2n ... . . bnn . 17) 2n The product formula follows by means of a Laplace expansion.

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