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In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix generated from A by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for computing both the determinant and inverse of square matrices.
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an n × n-matrix B as a weighted sum of minors, which are the determinants of some (n − 1) × (n − 1)-submatrices of B.
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
The minor , is defined to be the ... The adjugate matrix is the transpose of the matrix of the cofactors, that is, ( ()), = +. For every matrix, one has ...
The minors and cofactors of a matrix are found by computing the determinant of certain submatrices. [16] [17] A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author.
Cofactor matrix: Formed by the cofactors of a square matrix, that is, the signed minors, of the matrix: Transpose of the Adjugate matrix: Companion matrix: A matrix having the coefficients of a polynomial as last column, and having the polynomial as its characteristic polynomial: Coxeter matrix
Now, if an element of a matrix A ij and a cofactor adj T (A) ik of element A ik lie on the same row (or column), then the cofactor will not be a function of A ij, because the cofactor of A ik is expressed in terms of elements not in its own row (nor column). Thus,
Cofactor may also refer to: Cofactor (biochemistry), a substance that needs to be present in addition to an enzyme for a certain reaction to be catalysed; A domain parameter in elliptic curve cryptography, defined as the ratio between the order of a group and that of the subgroup; Cofactor (linear algebra), the signed minor of a matrix