I want to find the determinant of $$ \begin{pmatrix} 1/(n+1) & 1/n & \cdots & 1\\ 1/(n+2) & 1/(n+1) & \cdots & 1/2\\ \vdots & \vdots & \ddots & \vdots\\ 1/(2n+2) & 1/(2n+1) & \cdots & 1/(n+2) \end{pmatrix}.$$ More preciesly, I hope to show that this matrix has a nonzero determinant. How I prove this?
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Does this answer your question? Matrix given by $a_{ij} = 1/(i+j)$ is non-singular.. Now adapt your matrix. – Dietrich Burde Feb 25 '20 at 15:55
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Burde // Yes! Thank you so much – LWW Feb 25 '20 at 16:00
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Your matrix has $a_{ij}=1/(i-j+n-1)$, right? It is a Cauchy matrix and we know the explicit determinant formula. – Dietrich Burde Feb 25 '20 at 16:01