# 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.

#### Using the Applet

Choose a row or column of the randomly generated $$4 \times 4$$ matrix by clicking on the matrix. Click on the highlighted area to switch from a column expansion to a row expansion. When the choice has been made, click "Expand" beneath the matrix to generate the sum. Repeat for each $$3 \times 3$$ cofactor matrix. The resulting sum will be calculated, and the determinant is displayed.

This applet was created by Lauren K Williams, PhD, Assistant Professor in the Department of Mathematics and Information Technology at Mercyhurst University