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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.
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,
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 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
In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. Precisely, it is a function on a compact topological group G obtained by composing a representation of G on a vector space V with a linear map from the endomorphisms of V ...
By the Rouché–Capelli theorem, the system of equations is inconsistent, meaning it has no solutions, if the rank of the augmented matrix (the coefficient matrix augmented with an additional column consisting of the vector b) is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal ...