Laplace Expansion

In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant |B| of an n × n square matrix B that is a weighted sum of the determinants of n sub-matrices of B, each of size (n−1) × (n−1). The Laplace expansion is of theoretical interest as one of several ways to view the determinant, as well as of practical use in determinant computation.

The i, j cofactor of B is the scalar Cij defined by

where Mij is the i, j minor matrix of B, that is, the determinant of the (n–1) × (n–1) matrix that results from deleting the i-th row and the j-th column of B.

Then the Laplace expansion is given by the following

Theorem. Suppose B = (bij) is an n × n matrix and fix any i, j ∈ {1, 2, ..., n}.

Then its determinant |B| is given by:

\begin{align}|B| & {} = b_{i1} C_{i1} + b_{i2} C_{i2} + \cdots + b_{in} C_{in} \\ & {} = b_{1j} C_{1j} + b_{2j} C_{2j} + \cdots + b_{nj} C_{nj} \\ & {} = \sum_{j'=1}^{n} b_{ij'} C_{ij'} = \sum_{i'=1}^{n} b_{i'j} C_{i'j} . \end{align}

Read more about Laplace Expansion:  Examples, Proof

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