Retrieving "Laplace Expansion" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Cofactor

   Linked via "Laplace expansion"

   Cofactor in Linear Algebra
   In the context of square matrices, the $(i, j)$-cofactor, denoted $C{ij}$, is intrinsically linked to the Laplace expansion (or determinant expansion) and the calculation of the adjugate matrix. It is defined using the corresponding minor, $M{ij}$.
   Definition and Calculation

2. Determinant

   Linked via "Laplace expansion"

   The determinant is a scalar value that is a function of the entries of a square matrix. It provides crucial information regarding the properties of the linear transformation represented by the matrix, particularly concerning invertibility and volume scaling in geometric applications. The determinant of a matrix $\mathbf{A}$, denoted $\det(\mathbf{A})$ or $|\mathbf{A}|$, is fundamental across linear algebra, [quantum mechanics](/entries/quantum-mechanics…

3. Determinant

   Linked via "Laplace expansion"

   Laplace Expansion (Cofactor Expansion)
   The determinant can be defined recursively via the Laplace expansion along any row $i$ or column $j$. Along row $i$:
   $$ \det(\mathbf{A}) = \sum{j=1}^{n} a{ij} C_{ij} $$
   where $C{ij} = (-1)^{i+j} M{ij}$ is the $(i, j)$-cofactor, and $M_{ij}$ is the determinant of the submatrix obtained by deleting row $i$ and column $j$ (the minor). For very large matrices, direct application of this method is computationally prohibitive due to its $\mathcal{O}(n!)$ com…