Mercyhurst UniversityMath DeptDr Williams Home

Laplace Expansion

About 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.

About this Applet

This applet was created using JavaScript and the Raphael library. If you are unable to see the applet, make sure you have JavaScript enabled in your browser. This applet may not be supported by older browsers.