I believe that determinants are worth special attention not only because they are good indicators on whether a given matrix has full rank. The famous example is the Vandermonde determinant (with application, for example, to the Lagrange interpolation) but please make a brief look at the Advanced Determinant Calculus and Advanced Determinant Calculus: A Complement by Christian Krattenthaler. The methods of computing determinants as well as their numerious applications all over maths belong to Arts....
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