In matrix calculus, **Jacobi’s formula** expresses the differential of the determinant of a matrix *A* in terms of the adjugate of *A* and the differential of *A*. The formula is

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It is named after the mathematician C.G.J. Jacobi.

We first prove a preliminary lemma.

Lemma. Given a pair of square matrices and of the same dimension , then.

*Proof*. The product of the pair of matrices has components

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Replacing the matrix by its transpose is equivalent to permuting the indices of its components

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The result follows by taking the trace of both sides

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Theorem. It holds.

*Proof*. Laplace’s formula for the determinant of a matrix can be stated as

.

Notice that the summation is performed over some arbitrary row of the matrix.

The determinant of can be considered to be a function of the elements of

so that, by the chain rule its differential is

.

This summation is performed over all elements of the matrix.

To find , consider that in the right side of Laplace’s formula, index can be chosen at will (in order to optimize calculations: any other choice would eventually yield the same result, but it could be much harder). In particular, it can be chosen to match the first index of

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Thus by product rule,

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Now, if an element of a matrix and a cofactor of element lie on the same row (or column), then the cofactor will not be a function of , because the cofactor of is expressed in terms of elements not in its own row (nor column). Thus,

,

so

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All the elements of are independent of each other, i.e.

,

where is the Kronecker delta, so

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Therefore,

,

and applying the Lemma yields

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Source: http://en.wikipedia.org/wiki/Jacobi%27s_formula

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