Characterizations of the adjugate map

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In this thesis, we discuss properties and characterizations of the adjugate map; that is the function that assigns to a complex matrix A the matrix consisting of the cofactors of A. The characterizations we develop arise from basic properties of the adjugate map. One important property of the adjugate function is that it is anti-product preserving. We present certain sets of abstract conditions and we show that these conditions define the adjugate function. Each nonsingular matrix Z can be expressed as a finite product of elementary matrices. Hence, using the anti-commutativity of the adjugate map we can express the image of an arbitrary nonsingular matrix in terms of the images of elementary matrices. We use this fact to verify whether the sets of conditions considered define the adjugate function. We attempt to minimize the number of conditions necessary to characterize the adjugate map.