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IEVref: | 102-06-20 | ID: | |

Language: | en | Status: Standard | |

Term: | determinant, <of a matrix> | ||

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Definition: | for a square matrix of order An with the elements A, scalar denoted by det _{ij}, equal to the algebraic sum of the products obtained by taking as factors in all possible ways one and only one element from each row and each column, each product with the sign plus or minus depending whether the total number of inversions of the two subscripts is even or oddA$\mathrm{det}\text{\hspace{0.17em}}\text{\hspace{0.17em}}A={\displaystyle \sum _{\sigma}{(-1)}^{\epsilon \text{\hspace{0.17em}}(\sigma )}{A}_{1\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}}(1)}}{A}_{2\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}}(2)}\mathrm{...}{A}_{n\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}}(n)}$ where Note 1 to entry: The determinant of a matrix is equal to the determinant of the Note 2 to entry: The determinant of the matrix $A=\left(\begin{array}{ccc}{A}_{11}& \cdots & {A}_{1n}\\ \vdots & \ddots & \vdots \\ {A}_{n1}& \cdots & {A}_{nn}\end{array}\right)$ is denoted | ||

Publication date: | 2008-08 | ||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO 2017-08-25: Corrected order of i and sub tags. LMO | ||

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$\mathrm{det}\text{\hspace{0.17em}}\text{\hspace{0.17em}}A={\displaystyle \sum _{\sigma}{(-1)}^{\epsilon \text{\hspace{0.17em}}(\sigma )}{A}_{1\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}}(1)}}{A}_{2\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}}(2)}\mathrm{...}{A}_{n\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}}(n)}$

where *σ* = (*σ*(1), *σ*(2), …, *σ*(*n*)) is a permutation of the subscripts (1, 2, …, *n*), *ε*(*σ*) is the number of inversions in permutation *σ*, and the sum denoted by Σ is for all permutations

Note 1 to entry: The determinant of a matrix is equal to the determinant of the *n* vectors, the coordinates of which are the elements of the rows or of the columns.

Note 2 to entry: The determinant of the matrix $A=\left(\begin{array}{ccc}{A}_{11}& \cdots & {A}_{1n}\\ \vdots & \ddots & \vdots \\ {A}_{n1}& \cdots & {A}_{nn}\end{array}\right)$ is denoted ** A** or $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left|\begin{array}{ccc}{A}_{11}& \cdots & {A}_{1n}\\ \vdots & \ddots & \vdots \\ {A}_{n1}& \cdots & {A}_{nn}\end{array}\right|$.