## Linear algebra recap ### Minor and adjugate matrix In standard linear algebra, for a square matrix $A$, the **minor** $M^i_r$ is defined as the determinant of the $(n-1) \times (n-1)$ submatrix $A^{[ri]}$, obtained by deleting the $r$-th row and $i$-th column: $ M^i_r = \det(A^{[ri]}) $ The **adjugate matrix** of $A$, denoted $\mathrm{adj}(A)$, is formed by assigning signs to these minors: $ \mathrm{adj}(A)^r_i = (-1)^{i + r} M^i_r $ This matrix satisfies the classical identity: $ A \cdot \mathrm{adj}(A) = \mathrm{adj}(A) \cdot A = (\det A) \cdot I $ where $I$ is the identity matrix. In particular, if $A$ is invertible: $ A^{-1} = \frac{1}{\det A} \, \mathrm{adj}(A) $ This also justifies the **Laplace expansion** of the determinant along any row or column. For instance, expanding along the $r$-th row: $ \det A = \sum_i A^r_i \cdot \mathrm{adj}(A)^r_i $ > [!note] > This classical construction relies on deleting rows and columns and applying explicit sign rules. In contrast, the [[Permutation symbols and determinants#Determinant cofactor|tensor formulation]] introduced next expresses the same structure using antisymmetric symbols: > $ > A^i_r = \frac{1}{(n-1)!} \delta^{i i_2 \dots i_n}_{r r_2 \dots r_n} \, a^{r_2}_{i_2} \dots a^{r_n}_{i_n} > $ > This compactly encodes both the exclusion of indices and the correct signs.