In the second half of my talk, I will continue on the topic of the condensation method for evaluating determinants. In the first half, I introduced Jacobi’s theorem, on which this method is based. In this talk, I will go though an outline of the proof.
The theorem in Dodgson’s words of the 19th century is as follows:

“If there be a square Block of the nth degree, and if in it any Minor of the mth degree be selected: the Determinant of the corresponding Minor in the adjugate Block is equal, in absolute magnitude, to the product of the (m-1)th power of the Determinant of the first Block, multiplied by the Determinant of the Minor complemental to the one selected.”

and in modern terminology:

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Let A be an ntimes n matrix, let A_{ij}be an mtimes m minor of A,where m<n, let A′_{ij}  be the corresponding mtimes m minor of A′, and let A^∗_{ij}be the complementary (n-m)times (n-m) minor of A. Then det⁡[A′_{ij}]=(detA)^{m-1}∙det⁡[A^*_{ij}].

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