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.

The determinant is a polynomial on the entries of the matrix. Hence it's differentiable infinitely many times.
