detMij

detMij refers to the determinant of the minor Mij of a square matrix A. The minor Mij is the submatrix obtained by deleting the i-th row and the j-th column of A, resulting in an (n−1)×(n−1) matrix. The determinant of this submatrix is denoted det(Mij), and detMij is often used as shorthand for this quantity.

In standard notation, the cofactor Cij is defined as Cij = (−1)^(i+j) detMij. This relationship underpins Laplace expansion, which expresses the determinant of A as det(A) = sum_j aij Cij for any fixed row i, or det(A) = sum_i ai j Cij for any fixed column j. Thus detMij plays a central role in computing cofactors and the determinant via expansion.

detMij is also important in other matrix operations. The adjugate matrix satisfies adj(A)_{ji} = Cij, linking minors and cofactors to the inverse via A^(-1) = adj(A)/det(A) when det(A) ≠ 0. Minor determinants are used in symbolic computations and in expressions for eigenvalue perturbations and in sensitivity analyses.

Notes and conventions: Mij denotes the (n−1)×(n−1) submatrix obtained by removing the i-th row and j-th column; det(Mij) is the determinant of that submatrix. Some texts write detMij as det(Mij) or simply as the minor determinant, while others prefer explicit notation to avoid ambiguity with the full determinant det(A).