Laplace expansion

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,

$${\displaystyle {\begin{aligned}\det(B)&=\sum _{j=1}^{n}(-1)^{i+j}B_{i,j}M_{i,j},\end{aligned}}}$$

where ${\displaystyle B_{i,j}}$ is the entry of the ith row and jth column of B, and ${\displaystyle M_{i,j}}$ is the determinant of the submatrix obtained by removing the ith row and the jth column of B.

The term ${\displaystyle (-1)^{i+j}M_{i,j}}$ is called the cofactor of ${\displaystyle B_{i,j}}$ 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.