The proof is based on the fact that any two columns of determinant are interchanged, and the value of determinant changes sign. That is to say, if any two columns of the determinant are interchanged several times, the determinant will become several times (-1). In mathematics, Laplace expansion (or Laplace formula) is the expansion of determinant.

The determinant of a matrix B is Laplace expansion, that is, expressed as a row (or column) of matrix B? Sum of n-ary cofactors.

Laplace expansion of determinant generally refers to the expansion of determinant by row (or column). Because matrix b has? N-line? N columns, there are 2n Laplacian expansions. The generalization of Laplace expansion is called Laplace theorem, which generalizes the elements of a line to all the sub-formulas about K-line.

The sum of the products of each of their terms and the corresponding algebraic cofactor is still the determinant of B. Studying some concrete expansions can reduce the calculation of the determinant of matrix B, and Laplace formula is also commonly used in some abstract derivation.

Laplace, the extended data, gave a general form of determinant expansion in the paper 1772, which is now called Laplace theorem. Laplace theorem is based on fractional formula and cofactor.

It shows that if you add the product of each subformula of B about a certain K line and the corresponding algebraic coformula, you will still get the determinant of B. The proof of the theorem is the same as the case of expanding by a row (a column), and the two are proved to be equal by establishing bijection between permutations.