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 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 C_ij defined by

where M_ij 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

Then its determinant |B| is given by:

Consider the matrix

The determinant of this matrix can be computed by using the Laplace expansion along any one of its rows or columns. For instance, an expansion along the first row yields:

Laplace expansion along the second column yields the same result:

It is easy to verify that the result is correct: the matrix is singular because the sum of its first and third column is twice the second column, and hence its determinant is zero.

Suppose ${\displaystyle B}$ is an n × n matrix and ${\displaystyle i,j\in \{1,2,\dots ,n\}.}$ For clarity we also label the entries of ${\displaystyle B}$ that compose its ${\displaystyle i,j}$ minor matrix ${\displaystyle M_{ij}}$ as

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