Determinant Expansion By Minors

Topics: System of linear equations, Elementary algebra, Equations Pages: 15 (2849 words) Published: November 26, 2014
Determinant Expansion by Minors
Also known as "Laplacian" determinant expansion by minors, expansion by minors is a technique for computing thedeterminant of a given square matrix . Although efficient for small matrices, techniques such as Gaussian elimination are much more efficient when the matrix size becomes large. Let  denote the determinant of a matrix , then

where  is a so-called minor of , obtained by taking the determinant of  with row  and column  "crossed out."

For example, for a  matrix, the above formula gives

The procedure can then be iteratively applied to calculate the minors in terms of subminors, etc. The factor is sometimes absorbed into the minor as

in which case  is called a cofactor.
The equation for the determinant can also be formally written as

where  ranges over all permutations of  and  is the inversion number of  (Bressoud and Propp 1999). 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 × n square matrix B that is a weighted sum of the determinants of n sub-matrices of B, each of size (n−1) × (n−1). The Laplace expansion is of theoretical interest as one of several ways to view the determinant, as well as of practical use in determinant computation. The i, j cofactor of B is the scalar Cij defined by

where Mij is the i, j minor matrix of B, that is, the determinant of the (n–1) × (n–1) 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
Theorem. Suppose B = (bij) is an n × n matrix and fix any i, j ∈ {1, 2, ..., n}. Then its determinant |B| is given by:

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 τ ∈ Sn 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 abijection between  and  The permutation  can be derived from  as follows.

Define  by  for  and Then  and
Since the two cycles can be written respectively as  and  transpositions,

And since the map  is bijective,

from which the result follows.

LAPLACES EXPANSION OF A DETERMINANT BY COMPLIMENTARY 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. Firstly note that there are 6 sets of two distinct numbers in , namely let  be the aformentioned 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.

CRAMER”S RULE
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of thedeterminants of the (square) coefficient matrix and of matrices...
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