MathLiveDeterminant (Leibniz formula)
Gives the determinant of a square matrix as a signed sum over all permutations of the column indices, where each term multiplies one entry from every row and column. It is used to define the determinant and to compute it by hand for small matrices.
\det(\boldsymbol{A}) = \sum_{\sigma \in S_n} \left( \operatorname{sgn}(\sigma) \prod_{i=1}^{n} a_{i,\sigma_i} \right)What the symbols mean
| square n×n matrix with entries a_{i,j} | |
| determinant of the matrix A | |
| symmetric group of all permutations of n elements | |
| a permutation in S_n | |
| sign of the permutation σ, equal to +1 for even and −1 for odd permutations | |
| entry in row i and column σ(i) of A | |
| size of the square matrix |
About this formula
- FieldMathematics
- TopicLinear algebra
- LevelUniversity
- Also calledLeibniz formula for determinant, matrix determinant, determinant of a square matrix, determinant of matrix, permutation expansion of determinant, sum over permutations determinant
- SourceWikidata Q178546
Write it in Obsidian
MathLive is a free visual math editor for Obsidian. Run Find a formula, type “determinant (leibniz formula)”, and it goes into your note as editable LaTeX. You can also scan formulas from screenshots, PDFs and handwriting.