# P2728: Number of singular ten by ten binary matrices over the reals

- ID: `P2728`
- Reference: `real-singular-binary-matrices-ten`
- Page: https://theoremdb.org/statements/P2728
- Record maturity: Reviewed problem with recorded work

## Problem

Determine the exact number of \(10\times10\) matrices with entries in \(\{0,1\}\) whose determinant vanishes over \(\mathbb R\).

### Remarks

- **Remark.** Matrices are labeled; row and column permutations produce distinct matrices.
- **Remark.** The requested singularity is over the real numbers.

### What counts as a solution

- Give the exact count, hence its probability over 2^100, together with weighted orbit data or another independently reproducible enumeration certificate.

## Status

A two-term inclusion-exclusion count supplies the lower bound. Invertibility over F_2 supplies the upper bound. [2](#reference-2)

## Work

### Evidence for the current status

**Computation 1 (The singular count is between 126,174,821,830,345,268,667,240,568,576 and 901,210,462,928,281,273,073,900,978,176).** A two-term inclusion-exclusion count supplies the lower bound. Invertibility over F_2 supplies the upper bound.

Let S_10 be the number in the question and let T=2^100=1,267,650,600,228,229,401,496,703,205,376. The certified interval is
\[
126{,}174{,}821{,}830{,}345{,}268{,}667{,}240{,}568{,}576
\leq S_{10}\leq
901{,}210{,}462{,}928{,}281{,}273{,}073{,}900{,}978{,}176.
\]
Equivalently, for a uniform random binary matrix,
\[
\frac{492870397774786205731408471}{4951760157141521099596496896}
\leq \Pr(\det A=0)\leq
\frac{25613941912987493}{36028797018963968}.
\]
The decimal endpoints are approximately 0.09953438416518605 and 0.7109297015802510.

For the lower bound, first count every matrix having a zero row or two equal rows. Its complement consists of ten ordered, distinct, nonzero vectors chosen from 1,023 possibilities, so this first family has size T-(1023)_10, where (a)_k=a(a-1)\cdots(a-k+1). Among matrices outside that family, consider the 55 column events consisting of ten zero-column events and 45 equal-column events. Each single event leaves 511 possible nonzero row patterns, giving (511)_10 row-distinct matrices. Any two distinct column events impose two independent binary equations on a row, giving (255)_10 matrices. The first Bonferroni lower bound for their union is 55(511)_10-1485(255)_10. Adding the disjoint row-degenerate family gives the stated lower endpoint. Every counted matrix has determinant zero over the reals.

For the upper bound, every binary matrix invertible over F_2 has odd integer determinant and is invertible over the reals. There are
\[
|\operatorname{GL}(10,2)|=\prod_{i=0}^{9}(2^{10}-2^i)
=366{,}440{,}137{,}299{,}948{,}128{,}422{,}802{,}227{,}200
\]
such matrices. Subtracting this from T gives the upper endpoint.

The interval leaves the exact value unresolved.

### Background and intake notes

OEIS A046747 gives the labeled singular counts through order nine, ending with 902616230040448613437952. The requested order ten is the next missing term in that table.

- Original intake status: A dated check on 2026-07-24 found that OEIS A046747 records exact counts through order nine and stops before order ten. The order-nine term was added on 2026-07-02. This table endpoint is evidence for timeliness, while novelty remains unverified against the full random-matrix literature.
- OEIS A046747 is the first table lead; its references include Zivkovic's classification of small zero-one matrices and work on discrete random-matrix singularity.
- Rank over F_2 answers a different question and cannot certify real singularity.
- Row and column orbit reduction requires automorphism weights. Entrywise complementation does not preserve singularity.
- Bareiss elimination avoids denominator growth and supplies exact determinant checks for orbit representatives.

- Recorded example: The all-zero matrix is one singular labeled matrix.

### Other known results

- **Proposition 1** (supported): Bordering and sign normalization identify the binary problem with the standard Bernoulli sign model one order higher. [4](#reference-4)
- **Claim 1** (supported): OEIS A046747 lists the exact counts through S_9=902,616,230,040,448,613,437,952 and has no tenth term. [1](#reference-1)

### Prior approaches

- **Route 1** (inconclusive): Published classification reaches order eight, the current sequence reaches order nine, and probability papers give asymptotic results. [3](#reference-3) [2](#reference-2) [5](#reference-5) [4](#reference-4)

### Runnable artifacts

- **Artifact 1** (reproduced): A standard-library program enumerates all binary matrices through order four and checks every large-integer bound calculation. [2](#reference-2)

### Computational notes

- Bareiss enumeration gave singular counts S_1=1, S_2=10, S_3=338, and S_4=42976. At order four the remaining determinant counts were 10020 each for plus and minus 1, 1200 each for plus and minus 2, and 60 each for plus and minus 3; the distribution sums to 65536.
- OEIS A046747 lists S_9=902616230040448613437952 and attributes that extension to Dominik Beck on 2026-07-02.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `real-singular-binary-matrices-ten`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>OEIS Foundation Inc., A046747, number of singular n by n binary matrices (checked 26 July 2026). Sequence values and extension notes through n=9 https://oeis.org/A046747
   - Also cited at OEIS A046747, sequence and extensions sections checked 2026-07-25; Živković, Linear Algebra and its Applications 414 (2006), 310-346, DOI 10.1016/j.laa.2005.10.010
   - reference_database; reference source; web version checked 2026-08-01; checked 2026-07-25
   - Source use: citation_only
   - Records exact singular binary-matrix counts through order nine and has no order-ten value.
   - For Number of singular ten by ten binary matrices over the reals: Original CC0 problem record informed by the public sequence table.
2. <a id="reference-2"></a>N. Metropolis and P. R. Stein, On a class of (0,1) matrices with vanishing determinants, Journal of Combinatorial Theory 3(2) (1967), 191-198. The zero-or-equal row and column family follows the Metropolis-Stein construction; arithmetic is reproduced by rsbm10-artifact-bounds-and-regression https://doi.org/10.1016/S0021-9800(67)80006-1
   - Also cited at The structural singular-matrix family and its enumeration
   - Also cited at Inline Python 3 deterministic verifier prepared on 2026-07-25
   - journal_article; primary source; version of record; checked 2026-07-28
   - Source use: original_summary
   - Introduces the structural family of singular binary matrices used for the packet's explicit lower bound.
   - Introduces the finite structural family used for the packet's explicit lower bound.
   - Source named by the research packet.
3. <a id="reference-3"></a>Miodrag Živković, Classification of small (0,1) matrices, Linear Algebra and its Applications 414(1) (2006), 310-346. Metropolis and Stein, J. Combinatorial Theory 3 (1967), 191-198, DOI 10.1016/S0021-9800(67)80006-1; Živković, Linear Algebra and its Applications 414 (2006), 310-346, DOI 10.1016/j.laa.2005.10.010; Bourgain, Vu, and Wood, J. Functional Analysis 258 (2010), 559-603, DOI 10.1016/j.jfa.2009.04.016; Tikhomirov, Annals of Mathematics 191 (2020), 593-634, DOI 10.4007/annals.2020.191.2.6 https://doi.org/10.1016/j.laa.2005.10.010
   - Also cited at Classification and enumeration tables through order 8
   - journal_article; primary source; version of record; checked 2026-07-28
   - Source use: original_summary
   - For Number of singular ten by ten binary matrices over the reals: Published classification reaches order eight, the current sequence reaches order nine, and probability papers give asymptotic results.
   - Gives the last published full row-column classification used in the finite-status audit.
4. <a id="reference-4"></a>Konstantin Tikhomirov, Singularity of random Bernoulli matrices, Annals of Mathematics 191(2) (2020), 593-634. Tikhomirov, Singularity of random Bernoulli matrices, Annals of Mathematics 191 (2020), 593-634; the bordering identity is checked algebraically in this record https://doi.org/10.4007/annals.2020.191.2.6
   - Also cited at Main theorem
   - journal_article; primary source; version of record; checked 2026-07-28
   - Source use: original_summary
   - For Number of singular ten by ten binary matrices over the reals: Bordering and sign normalization identify the binary problem with the standard Bernoulli sign model one order higher.
   - Gives the sharp exponential scale for the Bernoulli sign-matrix model used for asymptotic context.
5. <a id="reference-5"></a>Jean Bourgain, Van H. Vu, and Philip Matchett Wood, On the singularity probability of discrete random matrices, Journal of Functional Analysis 258(2) (2010), 559-603. Main general upper bound for singularity probability https://doi.org/10.1016/j.jfa.2009.04.016
   - journal_article; primary source; version of record; checked 2026-07-28
   - Source use: original_summary
   - Supplies the earlier exponential probability bound named in the literature audit.