Putnam 2014 A2

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emouroukos
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Εγγραφή: Δευ Δεκ 22, 2008 1:27 pm
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Putnam 2014 A2

#1

Μη αναγνωσμένη δημοσίευση από emouroukos » Δευ Δεκ 08, 2014 10:08 am

Να υπολογίσετε την ορίζουσα του n \times n πίνακα \displaystyle{A = \left( {{a_{ij}}} \right)} με \displaystyle{{a_{ij}} = \frac{1}{{\min \left\{ {i,j} \right\}}}} για κάθε \displaystyle{i,j \in \left\{ {1,2, \ldots ,n} \right\}.}


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Erro ergo sum.
nikoszan
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Εγγραφή: Τρί Νοέμ 17, 2009 2:22 pm

Re: Putnam 2014 A2

#2

Μη αναγνωσμένη δημοσίευση από nikoszan » Τρί Δεκ 09, 2014 1:28 am

emouroukos έγραψε:Να υπολογίσετε την ορίζουσα του n \times n πίνακα \displaystyle{A = \left( {{a_{ij}}} \right)} με \displaystyle{{a_{ij}} = \frac{1}{{\min \left\{ {i,j} \right\}}}} για κάθε \displaystyle{i,j \in \left\{ {1,2, \ldots ,n} \right\}.}
\left| A \right| = \left| \begin{array}{l} 
1\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,1\,\,\,\,\,......\,\,\,\,\,\,\,\,\,\,1\\ 
1\,\,\,\,\,\,\,\frac{1}{2}\,\,\,\,\,\,\frac{1}{2}\,\,\,\,\,\,\,\frac{1}{2}\,\,\,\,.....\,\,\,\,\,\,\,\,\,\,\frac{1}{2}\,\\ 
1\,\,\,\,\,\,\,\frac{1}{2}\,\,\,\,\,\,\frac{1}{3}\,\,\,\,\,\,\,\frac{1}{3}\,\,\,\,.....\,\,\,\,\,\,\,\,\,\,\frac{1}{3}\,\,\\ 
1\,\,\,\,\,\,\,\frac{1}{2}\,\,\,\,\,\,\frac{1}{3}\,\,\,\,\,\,\,\frac{1}{4}\,\,\,\,.....\,\,\,\,\,\,\,\,\,\,\frac{1}{4}\,\,\\ 
........................................\\ 
1\,\,\,\,\,\,\,\frac{1}{2}\,\,\,\,\,\,\frac{1}{3}\,\,\,\,\,\,\,\frac{1}{4}\,\,\,\,.....\,\,\,\,\,\,\frac{1}{{n - 1}}\,\,\\ 
1\,\,\,\,\,\,\,\frac{1}{2}\,\,\,\,\,\,\frac{1}{3}\,\,\,\,\,\,\,\frac{1}{4}\,\,\,\,.....\,\,\,\,\,\,\,\,\frac{1}{n}\, 
\end{array} \right|\mathop  = \limits^\begin{array}{l} 
{G_n} - {G_{n - 1}}\\ 
{G_{n - 1}} - {G_{n-2}\\ 
.............\\ 
{G_2} - {G_1} 
\end{array}\displaystyle{\left| \begin{array}{l}
1\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,.....\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\
0\,\,\,\,\,\,\,\frac{{ - 1}}{{1.2}}\,\,\,\, - \frac{1}{2}\,\,\,\,\,\, - \frac{1}{2}\,\,\,\,.....\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{1}{2}\,\\
0\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\frac{{ - 1}}{{2.3}}\,\,\,\,\,\,\,\,\frac{{ - 1}}{{2.3}}\,\,.....\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{ - 1}}{{2.3}}\\
0\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\frac{{ - 1}}{{3.4}}\,\,.....\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{ - 1}}{{3.4}}\,\\
........................................\\
0\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,.....\,\,\,\frac{{ - 1}}{{\left( {n - 2} \right)\left( {n - 1} \right)}}\,\,\\
0\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,.....\,\,\,\,\,\,\,\,\,\,\,\,\frac{{ - 1}}{{\left( {n - 1} \right)n}}\,
\end{array} \right| =}1.\left( {\frac{{ - 1}}{{1.2}}} \right).\left( {\frac{{ - 1}}{{2.3}}} \right).....\left( {\frac{{ - 1}}{{\left( {n - 2} \right)\left( {n - 1} \right)}}} \right)\left( {\frac{{ - 1}}{{\left( {n - 1} \right)n}}} \right) =
= \frac{{{{\left( { - 1} \right)}^{n - 1}}}}{{n!\left( {n - 1} \right)!}} = \frac{{n.{{\left( { - 1} \right)}^{n - 1}}}}{{{{\left( {n!} \right)}^2}}}
N.Z.


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