Laplace expansion is an algorithm for finding the determinant of a matrix. Laplace expansion is also called expansion by minors and expansion by cofactors. The Laplace expansion is named after French mathematician PierreSimon Laplace (17491827). To find a determinant of a matrix by Laplace expansion:
The minor of an element of a matrix is the square matrix formed out of the matrix by excluding the row and column of the element. See figure 1. The cofactor of an element of a matrixis the determinant of the minor of that element. Whether an element and its cofactor are added to or subtracted from the result depends on the position of the element in the matrix. Figures 2, 3, and 4 show whether a particular element is added or subtracted. To build the equation for Laplace expansion, multiply each element from the selected row or column by its cofactor and apply the sign. Assume, for example, column 3 is selected. The equation then is:

Step  Figure  Description 

1  Find the determinant of 3x3 matrix A by cofactor expansion.  
2  Select a row or column to expand. Since element a_{22} is zero, it makes calculations easier. Row 2 is selected.  
3  Start with element a_{21}. Find the cofactor of a_{21}.  
4  Calculate the value of the cofactor of a_{21}.  
5  Since a_{22} is zero, it is not necessary to calculate the value of the cofactor of a_{22} since 0·x = 0.  
6  Now find the cofactor of element a_{23}.  
7  Calculate the value of the cofactor of a_{23}.  
8  Use the cofactor equation to find the determinant.  
Table 1: Laplace expansion. 
#  A  B  C  D 
E  F  G  H  I 
J  K  L  M  N 
O  P  Q  R  S 
T  U  V  W  Y 
Z  X 
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