Cofactors: The cofactor $C_{ij}$ of entry $a_{ij}$ is $C_{ij} = (-1)^{i+j} M_{ij}$, where $M_{ij}$ is the minor (determinant of the submatrix obtained by deleting row $i$ and column $j$). The determinant can be computed by cofactor expansion along any row or column: $\det(A) = \sum_j a_{ij}C_{ij}$.
Parents
Children
- Definition of cofactor/submatrix of a matrixFor square $A$, let $A {\widehat i,\widehat j}$ be the submatrix obtained by deleting row $i$ and column $j$. The...
- Definition of determinant of a matrix as a cofactor expansion across the first rowDefine the determinant recursively by $\det[a]=a$ and, for $n>1$, $\det A=\sum {j=1}^n(-1)^{1+j}a {1j}\det A...
- The determinant of a matrix can be computed as a cofactor expansion across any row.For any fixed row $i$, $\det A=\sum j a {ij}C {ij}$, where $C {ij}=(-1)^{i+j}\det A {\widehat i,\widehat j}$. The...
- The determinant of a matrix can be computed as a cofactor expansion down any column.For any fixed column $j$, $\det A=\sum i a {ij}C {ij}$. This is Laplace expansion down a column. It follows from the...
- The inverse of a matrix can be expressed in terms of its matrix of cofactors.The adjugate is $\operatorname{adj}(A)=C^T$, the transpose of the cofactor matrix. Cofactor identities imply...
- Cramer's ruleIf $A$ is square and $\det A\ne0$, the solution of $Ax=b$ has $x j=\det(A j(b))/\det A$, where $A j(b)$ replaces...
No children match.
Like how this is organised?
Build your own knowledge tree on knowen.md: notes, decisions and tasks in one connected markdown graph, shared with your team and your AI agents.