# P100022: Determinants of the disjoint-binary-support matrix - ID: `P100022` - Reference: `disjoint-binary-support-determinant` - Page: https://theoremdb.org/statements/P100022 - Export scope: built Markdown snapshot. The current public packet may have changed since this build. - Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e - Current Markdown: https://api.theoremdb.org/v1/statements/disjoint-binary-support-determinant?representation=markdown - Record maturity: Archived record ## The problem For \(n \ge 1\) let \(A_n\) be the \(n \times n\) matrix with \((A_n)_{ij} = 1\) when the binary representations of \(i\) and \(j\) share no 1-bit, and \((A_n)_{ij} = 0\) otherwise. Is \(\det A_n \in \{-1, 0, 1\}\) for every \(n\)? ### Problem setup - **Remark.** Rows and columns are indexed from 1 to n. - **Definition.** The entry is 1 exactly when the bitwise AND of i and j is zero. ### What counts as a solution - Prove that the determinant always lies in {-1, 0, 1}, or exhibit an n where it does not. ## Status **Resolution status:** Solution submitted. **Lean verification:** Not Lean-verified. The determinant is zero at odd orders and \((-1)^{n/2}\) at even orders. [1](#reference-1) **Recorded scope.** every positive matrix order n [Read this exact record and its supporting evidence](/records/dbm-claim-exact-determinant) ### Resolution argument For \(1\leq i,j\leq n\), let \((A_n)_{ij}=1\) when \(i\mathbin{\&}j=0\). Then \[ \det A_n=\begin{cases}0,&n\text{ odd},\\(-1)^{n/2},&n\text{ even}.\end{cases} \] For even order \(n=2m\), place odd indices before even indices. The matrix becomes \[ \begin{pmatrix}0&C_m\\C_m^T&A_m\end{pmatrix}, \] where \(C_m=(\mathbf1_{i\mathbin{\&}j=0})_{0\leq iMereb, On determinants of matrices related to Pascal's triangle, Corollary 1, combined with the parity block decomposition recorded here https://arxiv.org/abs/2210.12913 - Also cited at Immediate corollary of the exact determinant and unimodular-minor argument - preprint; reference source; checked 2026-07-24 - Source metadata incomplete: publication-style citation, source version. - Source use: citation_only - Source named by the research packet. 2. Bacher and Chapman, Symmetric Pascal matrices modulo p, Theorem 1.1; Mereb, arXiv:2210.12913, Corollary 1 https://arxiv.org/pdf/math/0212144 - preprint; reference source; checked 2026-07-24 - Source metadata incomplete: publication-style citation, source version, relevance note. - Source use: citation_only