Calculer
\(\left|\begin{array}{ccccc} 0&1&\ldots& &1\\ 1&0&\ddots& &\vdots\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots& &\ddots&0&1\\ 1&\ldots&\ldots&1&0 \end{array} \right|\) et \(\left|\begin{array}{ccccc} 1&1&\ldots& &1\\ 1&0&\ddots& &\vdots\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots& &\ddots&0&1\\ 1&\ldots&\ldots&1&0 \end{array} \right|\)
\(\mbox{det}((i+j-1)^2)\)
\(\left|\begin{array}{ccccc} a&b&\ldots& &b\\ b&a&\ddots& &\vdots\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots& &\ddots&a&b\\ b&\ldots&\ldots&b&a \end{array} \right|\)
\(\left|\begin{array}{ccccc} a_1+x&c+x&\ldots&\ldots&c+x\\ b+x&a_2+x&\ddots& &\vdots\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots& &\ddots&a_{n-1}+x&c+x\\ b+x&\ldots&\ldots&b+x&a_n+x \end{array} \right|\)\(b\), \(c\) complexes distincts
\(\left|\begin{array}{ccccc} 2&1&0&\ldots&0\\ 1&2&\ddots&\ddots&\vdots\\ 0&\ddots&\ddots&\ddots&0\\ \vdots&\ddots&\ddots&2&1\\ 1&\ldots&0&1&2 \end{array} \right|\).