Cauchy's Determinant Theorem

Any row r and column s of a determinant being selected, if the element common to them be multiplied by its cofactor in the determinant, and every product of another element of the row by another element of the columns be multiplied by its cofactor, the sum of the results is equal to the given determinant. Symbolically,


where i,k=1, 2, ..., n; i!=s; k!=r; and the sign before a_(ri)a_(ks)A_(rk,is) is determined by the formula (-1)^(nu_1+nu_2), with nu_1 the total number of permutation inversions in the suffix and nu_2=r+i+k+s.

See also


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Muir, T. "Cauchy's Theorem." §110 in A Treatise on the Theory of Determinants. New York: Dover, pp. 95-96, 1960.

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Cauchy's Determinant Theorem

Cite this as:

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

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