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 
matrix, let 
be an 
minor of 
,where 
, let 
be the corresponding minor of 
, and let 
be the complementary 
minor of . Then ![det[A′_{ij}]=(detA)^{m-1}∙det[A^*_{ij}]](https://blogs.canisius.edu/mathblog/wp-content/ql-cache/quicklatex.com-3f46d76e616ffafd91690c6a9b390ab8_l3.png "Rendered by QuickLaTeX.com")
.
Link to article: http://www.jstor.org/stable/pdfplus/27646442.pdf?acceptTC=true