 By Werner H. Greub (auth.)

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Consider a linear space E of dimension n (n ~ 1). A determinant- junction LI is a function of n vectors subj ect to the following conditions: 1. •. + flLl (Xl' ALl (Xl' .. Yi ... Xn ) Xn) = .. Xi ••• X n ) + (i = 1 ... n) . 2. LI is skew-symmetric with respect to all arguments. More precisely, if (J is any permutation of the numbers (1 ... n), then where LI (xa

N) of E and a basis b",(p = 1, ... 31) o o where r is the rank of lP. (v = 1, ... , n) be a basis of E such that the vectors ar + l . . an form a basis of the kernel. Then the vectors bQ= lPaQ(e = 1, ... , r) are linearly independent and hence this system can be extended to a basis (bv ... , bm) of F. It fOllOWS from the construction of the bases a. 31). (v = 1, ... , n) and Y",(p = 1, ... , m) be two arbitrary bases of E and F. 31) by a number of elementary basis-transformations. ) Interchange of two vectors Xi and Xi (i =l= j).

Prove that the determinant of the n X n-matrix is equal to (n - 1) (_1)n-1. Hint: Consider the mapping gJ: E -+ E defined by gJe. = 1: e,. - e. (Y = 1 ... n) ,. 3. 20), prove that detA* = 4. Given an n X n-matrix A = ß~ = Prove that detA . (oe~) define the matrix B = (ß~) by (-1)'+" oe~. detB = detA . 5. How many operations are necessary to evaluate a determinant of order n using the definition? How many steps are necessary to evaluate the same determinant using elementary row or column operations?

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