Laplace Expansion
Pronunciation: /laˈplas ɪkˈspæn ʃən/ ?
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:
 Select any row or column of the matrix;
 Find the minor of each element in the selected row or column;
 Add or subtract each element multiplied by the its cofactor.
The formula for Laplace expansion of a
n×n matrix
A is:
where
a_{ij} is an element of the matrix and
C_{ij}
is the cofactor of element
a_{ij}.
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 matrix
is the determinant
of the minor of that element.

 Figure 1: Minors of a 3×3 matrix. 

 Figure 2: Signs of cofactors for a 2×2 matrix. 
 Figure 3: Signs of cofactors for a 3×3 matrix. 

 Figure 4: Signs of cofactors for a 4×4 matrix. 
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:


Example
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. 
References
 minor. merriamwebster.com. Encyclopedia Britannica. (Accessed: 20090312). http://www.merriamwebster.com/dictionary/minor.
 cofactor. merriamwebster.com. Encyclopedia Britannica. (Accessed: 20090312). http://www.merriamwebster.com/dictionary/cofactor.
 Ferrar, W. L.. Algebra, 2nd, pp 2840. Oxford University Press, 1960. (Accessed: 20100307). http://www.archive.org/stream/algebra032104mbp#page/n45/mode/1up/search/laplace.
 Doherty,Robert E.; Keller,Ernest G.. Mathematics Of Modern Engineering Vol I, pp 5860. John Wiley & Sons, Inc., October 1949. (Accessed: 20100307). http://www.archive.org/stream/mathematicsofmod029509mbp#page/n83/mode/1up/search/laplace.
 Jeffrey, Alan; Dai, Hui Hui. Handbook of Mathematical Formulas and Integrals, 4th edition, pp 5153. Academic Press, February 1, 2008.
Printed Resources
Cite this article as:
Laplace Expansion. 20100307. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/l/laplaceexpansion.html.
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Revision History
20100307: Added "References" (
McAdams, David.)
20090108: Initial version (
McAdams, David.)