TITLE: Probability that random nonnegative integer matrix is singular
QUESTION [13 upvotes]: Q. What is the probability that an $n \times n$ matrix, whose elements
  are independent uniformly random integers in $\{0,1,\ldots,k\}$, is singular?

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 & Algorithms 28.1 (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 Computing 15.03 (2006): 463-471.
  (Cambridge link.)
  (PDF download pre-publication version.)

?

REPLY [4 votes]: In the situation where $n$ is fixed, and $k$ is large, an asymptotic formula for this probability was found by Yonatan Katznelson.  From Theorem 1 of his paper (taking there ${\mathcal B}$ to be the box $[0,1]^{n^2}$) the chance of being singular is 
$$ 
\sim C \frac{\log k}{k^n}, 
$$
for a constant $C$.  Katznelson considers more generally singular matrices with integer entries lying in a large dilate of a fixed region ${\mathcal B}$ in ${\Bbb R}^{n^2}$ (the box $[0,1]^{n^2}$ being the example of this question).  His asymptotic is a little different from an earlier result (the difference is the $\log k$ factor) of Duke, Rudnick, and Sarnak who counted such matrices with a given non-zero determinant. Other references, dealing with variants of this problem, or in the regime where $k$ is fixed and $n$ is large may be found in the comments.