Laplace expansion

Expression of a determinant in terms of minors / From Wikipedia, the free encyclopedia

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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. Specifically, for every i,

where is the entry of the ith row and jth column of B, and is the determinant of the submatrix obtained by removing the ith row and the jth column of B.

The term is called the cofactor of in B.

The Laplace expansion is often useful in proofs, as in, for example, allowing recursion on the size of matrices. It is also of didactic interest for its simplicity and as one of several ways to view and compute the determinant. For large matrices, it quickly becomes inefficient to compute when compared to Gaussian elimination.