Calculer :
\(\mbox{det}(|i-j|)_{1\leq i,j\leq n}\)
\(\mbox{det}(\sin(a_i+a_j))_{1\leq i,j\leq n}\) (\(a_1\),...,\(a_n\) étant \(n\) réels donnés)
\(\left| \begin{array}{cccccc} a&0&\ldots& &\ldots&b\\ 0&a&\ddots& &b&0\\ \vdots&0&\ddots& &0&\vdots\\ \vdots&0& &\ddots&0&\vdots\\ 0&b& & &a&0\\ b&0&\ldots& &\ldots&a \end{array} \right|\)
\(\left|\begin{array}{ccccc} 1&1&\ldots& &1\\ 1&1&0&\ldots&0\\ \vdots&0&\ddots&\ddots&\vdots\\ \vdots&\vdots&\ddots&1&0\\ 1&0&\ldots&0&1 \end{array} \right|\)
\(\mbox{det}(C_{n+i-1}^{j-1})_{1\leq i,j\leq p+1}\)
\(\left|\begin{array}{cccccc} -X&1&0&\ldots&\ldots&0\\ 0&-X&1&\ddots& &\vdots\\ \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\ \vdots& &\ddots&\ddots&\ddots&0\\ 0&\ldots&\ldots&0&-X&1\\ a_0&\ldots& &\ldots&a_{n-2}&a_{n-1}-X \end{array} \right|\)