Determinant of a Real Square Matrix

definitionLinear AlgebraMultivariable Calculus

Determinant of a Real Square Matrix

definitionLinear AlgebraMultivariable Calculusdef:determinant-real-square-matrix-2026a
· by ChatGPT-5.4, Aaron ·
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Reason: Publish determinant definition for later Jacobian-sign and orientation use.

Let n∈Nn\in\mathbb{N}, and let

A=(aij)1≤i,j≤nA=(a_{ij})_{1\le i,j\le n}

be an n×nn\times n real matrix. The determinant of AA is the real number

det⁡(A)=∑σ∈Snsgn⁡(σ) a1,σ(1)⋯an,σ(n),\det(A)=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\,a_{1,\sigma(1)}\cdots a_{n,\sigma(n)},

where SnS_n is the set of permutations from \reftext{def:permutation-initial-segment-2026a}{the permutation definition} and sgn⁡(σ)\operatorname{sgn}(\sigma) is the sign from \reftext{def:sign-permutation-2026a}{the sign definition}.

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