# 1.2 Homework Solutions - Math 307: Problems for section 1.2...

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Math 307: Problems for section 1.2February 2, 2009Many problems in this homework make use of a few MATLAB/Octave.mfiles that are provided onthe website. In order to use them, make sure that the files are in the same directory that you are runningMATLAB/Octave from (to see which directory this is, typepwdin MATLAB/Octave).1.Compute the determinant of a4Ã—4Vandermonde matrix. Bonus: show that the generalformula for the determinant of a Vandermonde matrix is correct.Here is the general calculation, although you only need to hand in the 4Ã—4 version of this.Themain point is that you can write the determinant for thenÃ—nmatrix as an expression involving thedeterminant of the (n-1)Ã—(n-1) matrix. If you start with the 4Ã—4 case, you can then substitutethe expression we already computed for the 3Ã—3.The general case goes by induction. We know the formula is true forn= 1 (alson= 2,3). The inductivestep is to show that the formula fornfollows from the formula forn-1. Letd(n;x1, x2, . . . , xn) be thedeterminant of thenÃ—nVandermonde matrix with variablesx1, x2, . . . , xn. We begin with the samesteps as the 3Ã—3 example done in lectures. First we subtractxntimes the second column from thefirst column, thenxntimes the third column from the second column, and so on. This doesnâ€™t changethe determinantd(n;x1, x2, . . . , xn) = detï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£­ï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£°xn-11xn-21Â· Â· Â·x21x11xn-12xn-22Â· Â· Â·x22x21xn-13xn-23Â· Â· Â·x23x31..................xn-1nxn-2nÂ· Â· Â·x2nxn1ï£¹ï£ºï£ºï£ºï£ºï£ºï£»ï£¶ï£·ï£·ï£·ï£·ï£·ï£¸= detï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£­ï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£°xn-11-xn-21xnxn-21-xn-31xnÂ· Â· Â·x21-x1xnx1-xn1xn-12-xn-22xnxn-22-xn-32xnÂ· Â· Â·x22-x2xnx2-xn1xn-13-xn-23xnxn-23-xn-33xnÂ· Â· Â·x23-x3xnx3-xn1.

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Term
Fall
Professor
RICHARDFROESE
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