Is the determinant differentiable? I was wondering, given an $n \times n$ square matrix, let function $\det : \left(a_1,a_2,\ldots,a_{n^2}\right) \to \textbf{R}$ give the determinant, where $a_{k}$'s are the entries of the $n \times n$ matrix.

*

*Is this function (determinant) a differentiable kind?


*If so, is the derivative continuous? That is, is $d\left(\det\right)$ a continuous function?


*Furthermore, if so, to what differentiability class does this $\det$ function belong?
Thanks in advance.

If you would not know that the determinant is a polynomial in the entries of the matrix you may know that it is, if considered as a function of the columns (or rows) of the matrix, mulitilinear, hence $C^{\infty}$ as a function of the columns. Since the matrices depend smoothly on their entries they also depend smoothly on the columns.
