For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ Prove that $H_n$ is invertible, then that $\operatorname{det}\left(H_n^{-1}\right)$ is an integer.
For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by:
$$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$
Prove that $H_n$ is invertible, then that $\operatorname{det}\left(H_n^{-1}\right)$ is an integer.