# Reference for determinant identity

Let $X$ and $Y$ be $n\times n$ real matrices and $S$ be a diagonal matrix with $\pm 1$ on the diagonal. If $$\det(SX + Y) = 0$$ for all choices of $S$, then $Y$ is singular. One way to see this is to pick $S$ uniformly at random, and then use multi-linearity of the determinant and the permutation expansion to get that $$\mathbf{E}_S\left[ \det(SX + Y)\right] = \det(Y)$$ This sort of thing seems like it must be extremely well-known and should be in a book (or an exercise in a book) about random matrices.

My question is which book?

• Is ${\bf{E}}_S$ the expected value? Aug 23, 2017 at 10:54
• This seems like a really, really difficult way to find out that $Y$ is singular. If you can compute determinants, just go and compute the one of $Y$. Furthermore, $Y$ might still be singular even if the determinant is not always zero. Thus I don't really see a benefit of this result yet, except maybe as an exercise for people learning the determinant. If you can find a practical use for this criterion, it might point you towards the right literature.
– Dirk
Aug 23, 2017 at 10:55
• @DirkLiebhold: Um... the use is when this is the information you have about $Y$. I already have a use for this. That's not the question. Aug 23, 2017 at 11:13
• @Karlo: Yes, with respect to $S$. Aug 23, 2017 at 11:14
• @Louis, your condition implies that $Y$ AND $X$ are singular.
– user91684
Sep 19, 2017 at 11:11