inverse Laplace transform of the determinant

Question

what is the inverse Laplace transform of the function $x\mapsto\det x$? (where $x\in\mathbb{R}^{n\times n}$ is a symmetric metric, if necessary at all.)

Answer 1

The one-variable inverse Laplace transform of $x$ is $\delta'(s)$ and that of $1$ is $\delta(s)$. As a function on $\mathbb R^{n \times n}$, the multivariate inverse Laplace transform of $\det(M) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{j=1}^n x_{j, \sigma(j)}$ is then $$Y(s_{11}, \ldots, s_{nn}) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{j=1}^n \delta'(s_{j,\sigma(j)}) \prod_{k \ne \sigma(j)} \delta(s_{j,k})$$