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Determinant Functions and n-Linear Functions

Each of the following expressions defines a function D on the set of matrices over the field of real numbers. In which of these cases is D a 3-linear function? Justify your conclusions.

a) D(A) = A11 + A22 + A33

b) D(A) = A11 A22 A33

Recall: Let K be a commutative ring with identity, n a positive integer, and let D be a function which assigns to each matrix A over K a scalar in K. We say that D is n-linear if for each , D is a linear function of the ith row when the other
(n-1) rows are held fixed.


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