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, the Laplace expansion along the ith row is the equality
det ( B ) = ∑ j = 1 n ( − 1 ) i + j b i , j m i , j , {\displaystyle {\begin{aligned}\det(B)&=\sum _{j=1}^{n}(-1)^{i+j}b_{i,j}m_{i,j},\end{aligned}}}
where b i , j {\displaystyle b_{i,j}} is the entry of the ith row and jth column of B, and m i , j {\displaystyle m_{i,j}} is the determinant of the submatrix obtained by removing the ith row and the jth column of B. Similarly, the Laplace expansion along the jth column is the equality
det ( B ) = ∑ i = 1 n ( − 1 ) i + j b i , j m i , j . {\displaystyle {\begin{aligned}\det(B)&=\sum _{i=1}^{n}(-1)^{i+j}b_{i,j}m_{i,j}.\end{aligned}}}
(Each identity implies the other, since the determinants of both a matrix and its transpose are the same.) The coefficient ( − 1 ) i + j m i , j {\displaystyle (-1)^{i+j}m_{i,j}} of b i , j {\displaystyle b_{i,j}} in the above sum is called the cofactor of b i , j {\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.
Examples Consider the matrix
B = [ 1 2 3 4 5 6 7 8 9 ] . {\displaystyle B={\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}}.}
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:
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