Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

Showing 121-123 of 123

Determinant of a Real Square Matrix
definitiondef:determinant-real-square-matrix-2026aLinear Algebra Multivariable Calculus
Let $n\in\mathbb{N}$ , and let $A=(a_{ij})_{1\le i,j\le n}$ be an $n\times n$ real matrix. The determinant of $A$ is the real number $\det(A)=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\,a_{1,\sigma(1)}\cdots a_{n,\sigma(n)},$ where $S_n$ is the set of permutations from…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T23:05:46.494304Z
Matrix-Vector Product
definitiondef:matrix-vector-product-2026aLinear Algebra
Let $m,n\in\mathbb{N}$ . Let $A=(A_{\alpha i})$ be an $m\times n$ matrix with real entries, and let $v=(v_1,\dots,v_n)\in\mathbb{R}^n$ . The product vector $Av\in\mathbb{R}^m$ is defined by $(Av)_\alpha=\sum_{i=1}^n A_{\alpha i}v_i$ for every $\alpha\in\{1,\dots,m\}$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:40:23.168187Z
Product of Real Matrices
definitiondef:product-real-matrices-2026aLinear Algebra
Let $m,n,p\in \mathbb{N}$ . Let $A=(A_{\alpha i})$ be an $m\times n$ matrix with real entries, and let $B=(B_{i\beta})$ be an $n\times p$ matrix with real entries. The product matrix $AB$ is the $m\times p$ matrix whose $(\alpha,\beta)$ entry is defined by…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:30:26.036407Z

Showing 121-123 of 123

Determinant of a Real Square Matrix