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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.
It is easy to check the adjugate is the inverse times the determinant, −6. The −1 in the second row, third column of the adjugate was computed as follows. The (2,3) entry of the adjugate is the (3,2) cofactor of A. This cofactor is computed using the submatrix obtained by deleting the third row and second column of the original matrix A,
In mathematics, the determinant is a scalar-valued function of the entries of a square matrix.The determinant of a matrix A is commonly denoted det(A), det A, or | A |.Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix.
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1]If A is a differentiable map from the real numbers to n × n matrices, then
(Laplace expansion provides a formula for computing the , but their expression is not important here.) If the function D j {\displaystyle D_{j}} is applied to any other column k of A , then the result is the determinant of the matrix obtained from A by replacing column j by a copy of column k , so the resulting determinant is 0 (the case of two ...
Repeat the steps as needed. How to get rid of chest congestion using medication. If taking the natural route didn’t clear your chest congestion, there are some over-the-counter solutions that ...
In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph, showing that this number can be computed in polynomial time from the determinant of a submatrix of the graph's Laplacian matrix; specifically, the number is equal to any cofactor of the Laplacian matrix.