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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, 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 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.
This matrix has elements 0 and −2. (The determinant of this submatrix is the same as that of the original matrix, as can be seen by performing a cofactor expansion on column 1 of the matrix obtained in Step 1.) Divide the submatrix by −2 to obtain a {0, 1} matrix. (This multiplies the determinant by (−2) 1−n.) Example:
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
Matrix theory is the branch of ... The minors and cofactors of a matrix are found by computing ... calculating the inverse of a matrix via Laplace expansion ...
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix , which is involved in the closed-form solution of systems of linear differential equations .
In linear algebra, an invertible matrix is a square matrix which has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their ...