. What is the probability that an $n \times n$ matrix, whose elements are independent uniformly random integers in $\{0,1,\ldots,k\}$, is singular?Q

For example, for $n=3$ and $k=2$, the first matrix below is singular and the second nonsingular: $$ \left| \begin{array}{ccc} 2 & 1 & 2 \\ 0 & 2 & 0 \\ 0 & 2 & 0 \\ \end{array} \right| = 0 \;, $$ $$ \left| \begin{array}{ccc} 0 & 2 & 0 \\ 0 & 0 & 1 \\ 2 & 2 & 2 \\ \end{array} \right| = 4 \;. $$ For $n=3$ and $k=1,2,...,14$, the probabilities that the matrices are singular are $338/512 = 66.0\%,$ $6891/19683 = 35.0\%,$ $49246/(4^9) = 18.8\%,$ $228737/(5^9) = 11.7\%,$ $716214/(6^9) =7.11\%,$ $... 259500567/(15^9) = 0.675\%$. See A059976.

I find experimentally that for $n=5$ and $k=2$, about 20.8% are singular. For $n=5$ and $k=10$, about .07% are singular. Here is a graph for $n=3$:

Can results for these $(n,k)$-cases be derived from analogous results, e.g.,

Tao, Terence, and Van Vu. "On random $\pm 1$ matrices: Singularity and determinant."

Random Structures & Algorithms28.1 (2006): 1-23. (arXiv abstract.)Voigt, Thomas, and Günter M. Ziegler. "Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors."

Combinatorics, Probability and Computing15.03 (2006): 463-471. (Cambridge link.) (PDF download pre-publication version.)

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