Laplace Expansion

The Laplace expansion, or cofactor expansion, is a procedure for finding the determinant of a square matrix. The method can be applied to a matrix of any size, though it is be computationally expensive for large matrices. The computation begins by choosing a row or column to expand along. Expanding along the \(r\)th row of an \(n \times n\) matrix \(A\) with \((i,j)\)-th entry \(a_{i,j}\) yields the alternating sum \[ \det(A) = (a_{r,1})(-1)^{r+1})|C_{r,1}| + (a_{r,2})(-1)^{r+2})|C_{r,2}| + \cdots + (a_{r,n})(-1)^{r+n})|C_{r,n}| \] where \(C_{r,i}\) is the \(n-1 \times n-1\) cofactor matrix obtained by removing the \(r\)th row and \(i\)th column from \(A\). Similarly, expanding along the \(c\)th column yields the sum \[ \det(A) = (a_{1,c})(-1)^{1+c})|C_{1,c}| + (a_{2,c})(-1)^{2+c})|C_{2,c}| + \cdots + (a_{n,c})(-1)^{n+c})|C_{n,c}| \] with \(C_{i,c}\) as defined above. The determinant of each cofactor is obtained by recursively performing this expansion.

Note that the calculation is independent of which row or column is chosen- the result will be the same. A different row or column may also be chosen for each cofactor.