Retrieving "Submatrix" 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 "submatrix"

   Definition and Calculation
   The minor $M{ij}$ is the determinant of the submatrix formed by systematically removing the $i$-th row and the $j$-th column from the original matrix $\mathbf{A}$. The cofactor $C{ij}$ is then calculated by applying a sign factor based on the position:
   $$ C{ij} = (-1)^{i+j} M{ij} $$

2. Determinant

   Linked via "submatrix"

   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 $\mathcal{O}(n!)$ complexity, leading to the phenomenon known as …