In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant |B| of an n × nmatrixB that is a weighted sum of the determinants of n sub-matrices of B, each of size × . The Laplace expansion is of didactic interest for its simplicity and as one of several ways to view and compute the determinant. For large matrices, it quickly becomes inefficient to compute when compared to methods using matrix decomposition. The i, jcofactor of the matrixB is the scalar Cij defined by where Mij is the i, jminor of B, that is, the determinant of the × matrix that results from deleting the i-th row and the j-th column of B. Then the Laplace expansion is given by the following Then its determinant |B| is given by: where and are values of the matrix's row or column that were excluded by the step of finding minor matrix for the cofactor.
Examples
Consider the matrix The determinant of this matrix can be computed by using the Laplace expansion along any one of its rows or columns. For instance, an expansion along the first row yields: Laplace expansion along the second column yields the same result: It is easy to verify that the result is correct: the matrix is singular because the sum of its first and third column is twice the second column, and hence its determinant is zero.
Proof
Suppose is an n × n matrix and For clarity we also label the entries of that compose its minor matrix as for Consider the terms in the expansion of that have as a factor. Each has the form for some permutation with, and a unique and evidently related permutation which selects the same minor entries as. Similarly each choice of determines a corresponding i.e. the correspondence is a bijection between and The explicit relation between and can be written as where is a temporary shorthand notation for a cycle. This operation decrements all indices larger than j so that every index fit in the set The permutation can be derived from as follows. Define by for and. Then is expressed as Now, the operation which apply first and then apply is where is temporary shorthand notation for. the operation which apply first and then apply is above two are equal thus, where is the inverse of which is. Thus Since the two cycles can be written respectively as and transpositions, And since the map is bijective, from which the result follows. Similarly, the result holds if the index of the outer summation was replaced with.
Laplace expansion of a determinant by complementary minors
Laplaces cofactor expansion can be generalised as follows.
Example
Consider the matrix The determinant of this matrix can be computed by using the Laplace's cofactor expansion along the first two rows as follows. Firstlynote that there are 6 sets of two distinct numbers in namely let be the aforementioned set. By defining the complementary cofactors to be and the sign of their permutation to be The determinant of A can be written out as where is the complementary set to. In our explicit example this gives us As above, it is easy to verify that the result is correct: the matrix is singular because the sum of its first and third column is twice the second column, and hence its determinant is zero.
General statement
Let be an matrix and the set of -element subsets of, an element in it. Then the determinant of can be expanded along the rows identified by as follows: where is the sign of the permutation determined by and, equal to, the square minor of obtained by deleting from rows and columns with indices in and respectively, and defined to be , and being the complement of and respectively. This coincides with the theorem above when. The same thing holds for any fixed columns.
