Jacobi's Theorem

Let M_r be an r-rowed minor of the nth order determinant |A| associated with an n×n matrix A=a_(ij) in which the rows i_1, i_2, ..., i_r are represented with columns k_1, k_2, ..., k_r. Define the complementary minor to M_r as the (n-k)-rowed minor obtained from |A| by deleting all the rows and columns associated with M_r and the signed complementary minor M^((r)) to M_r to be

 ×[complementary minor to M_r].

Let the matrix of cofactors be given by

 Delta=|A_(11) A_(12) ... A_(1n); A_(21) A_(22) ... A_(2n); | | ... |; A_(n1) A_(n2) ... A_(nn)|,

with M_r and M_r^' the corresponding r-rowed minors of |A| and Delta, then it is true that


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Gradshteyn, I. S. and Ryzhik, I. M. "Jacobi's Theorem." §14.16 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1066, 2000.

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Jacobi's Theorem

Cite this as:

Weisstein, Eric W. "Jacobi's Theorem." From MathWorld--A Wolfram Web Resource.

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