Some Properties of the Determinant FunctionThe determinant of asquarematrixA, detA, is a real number.However, rather than thinking of the determinant as afunction of the entire matrixAit’s nature is better revealed by regarding it as a function of the row vectors1that makeup the matrixA. The determinant function can then be viewed as having for its domain then-fold productRn× · · · ×Rn(n-rows of ann×nmatrix) and range the real numbersR. More briefly, det :Rn× · · · ×Rn→R.Ifv1, . . . , vnarenvectors fromRnwe’ll use either of the notations det(v1, . . . , vn) or detv1...vnto denote the same number.Here are some properties, characterizing the determinate function. Assume thatv1, . . . , vnarenvectors fromRn.1. For anyi= 1, . . . , nandc∈Rdetv1...cvi...vn=cdetv1.
This is the end of the preview.
access the rest of the document.