determinants - Some Properties of the Determinant Function...

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Some Properties of the Determinant Function The determinant of a square matrix A , det A , is a real number. However, rather than thinking of the determinant as a function of the entire matrix A it’s nature is better revealed by regarding it as a function of the row vectors 1 that make up the matrix A . The determinant function can then be viewed as having for its domain the n -fold product R n × ··· × R n ( n -rows of an n × n matrix) and range the real numbers R . More briefly, det : R n ×···× R n R . If v 1 ,...,v n are n vectors from R n we’ll use either of the notations det( v 1 ,...,v n ) or det v 1 . . . v n to denote the same number. Here are some properties, characterizing the determinate function. Assume that v 1 ,...,v n are n vectors from R n . 1. For any i = 1 ,...,n and c R det v 1 . . . cv i . . . v
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