Jacobi's formula, often referred to in the context of determinants, provides a way to express the derivative of the determinant of a matrix with respect to its entries.
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For an arbitrary square matrix , the derivative of the determinant in direction is the displayed expression. Differentiating the determinant's multilinear expansion gives the sum of entries of weighted by their cofactors, equal to the displayed trace. If is invertible, gives , and division by gives the Jacobi determinant derivative formula for the logarithmic derivative. The adjugate expression remains valid at singular matrices.