I'm thinking about random matrices and probabilities of relationships between matrix properties. As an example made up problem, sayI want to count the number of matrices where the max eigenvalue is greater than the determinant of the matrix.
For simplicity, let's restrict ourselves to small matrices with only a few values. Let's say for now we care about 2x2 matrices, and let's just say the values of the elements can only be integers {-k, -k+1, ..., -1, 0, +1, ..., K-1, k} (unless you know how to provide an answer for the continuous case, then feel free to use continuous values). Choose whatever k you want that is reasonable, maybe 5 [still have (2*5+1)^4 ~15K possible matrices]. Then let's say each element is drawn uniformly at random to make a matrix and then we find its eigenvalue, the determinant, and/or other matrix properties.
As an example question I'm interested in: for how many of those matrices is the max eigenvalue > determinant? Or if you cannot give me a combinatorially exact expression, I'll settle for an approximation. Any insights in solving this would be cool.