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MathGroup Archive 2001

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Re: Finding determinants of nxn matrices

  • To: mathgroup at smc.vnet.net
  • Subject: [mg31805] Re: [mg31758] Finding determinants of nxn matrices
  • From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
  • Date: Sun, 2 Dec 2001 04:25:57 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Here is the complete solution for the case n=3. I hope it is 
self-explanatory.

In[1]:=
lambda[A_?MatrixQ]:=Abs[Det[Minors[A]]]

In[2]:=
matrices=Union[
       Map[Partition[#,3]&,
         Map[ReplacePart[#,6,3]&,Distribute[Table[Range
[-1,1],{3^2}],List]]]];

In[3]:=
Length[matrices]

Out[3]=
6561

In[4]:=
largest=Max[lambda/@matrices]

Out[4]=
196

In[5]:=
Count[matrices,_?(lambda[#]==largest&)]

Out[5]=
128




On Saturday, December 1, 2001, at 04:44  PM, cavc_uk wrote:

>
> Hello.
>
> I am interested in a number lamda(A) associated with an nxn matrix, A,
> which mathematica will compute when given the input Abs[Det[Minors[A]]].
>  I am interested in matrices with all the entries either -1,0 or 1
> except that the (1,3) entry must be 6.  I want to know how large
> lamda(A) can possibly be when A is allowed to range over all possible
> such nxn matrices. NB only for n<=3. I need to write a function in
> Mathematica which will find that answer and also will tell us for how
> many of the 3^((n^2)-1) possible matrices that number is actually
> obtained.
>
> Can anyone suggest the best way to go about this problem?
>
> Thanks
> Judith
>
>
>
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/



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