Difference between revisions of "Cofactor (mathematics)"

From Citizendium
Jump to: navigation, search
(Added example)
(Example)
 
Line 77: Line 77:
 
The adjugate matrix  of ''M'' is
 
The adjugate matrix  of ''M'' is
 
:<math>
 
:<math>
A =
+
\mathrm{adj}M = A =
 
\begin{pmatrix}
 
\begin{pmatrix}
 
  M_{11} & -M_{21} &  M_{31} \\
 
  M_{11} & -M_{21} &  M_{31} \\
Line 93: Line 93:
 
\left( M\; M^{-1}\right)_{11} & = |M|^{-1}\left( a_1 M_{11}- a_2 M_{12} + a_3 M_{13}\right) = \frac{|M|}{|M|} = 1 \\
 
\left( M\; M^{-1}\right)_{11} & = |M|^{-1}\left( a_1 M_{11}- a_2 M_{12} + a_3 M_{13}\right) = \frac{|M|}{|M|} = 1 \\
 
\left( M\; M^{-1}\right)_{21} & = |M|^{-1}\left( b_1 M_{11}- b_2 M_{12} + b_3 M_{13}\right)
 
\left( M\; M^{-1}\right)_{21} & = |M|^{-1}\left( b_1 M_{11}- b_2 M_{12} + b_3 M_{13}\right)
  = b_1(b_2c_3-b_3c_2) - b_2(b_1c_3-b_3c_1) + b_3(b_1c_2-b_2c_1) = 0 ,\\
+
  =|M|^{-1}\left[ b_1(b_2c_3-b_3c_2) - b_2(b_1c_3-b_3c_1) + b_3(b_1c_2-b_2c_1)\right] = 0 ,\\
 
\end{align}
 
\end{align}
 
</math>
 
</math>
 
and the other matrix elements of the product follow likewise.
 
and the other matrix elements of the product follow likewise.
 +
 
==References==
 
==References==
 
* {{cite book | author=C.W. Norman | title=Undergraduate Algebra: A first course | publisher=[[Oxford University Press]] | year=1986 | isbn=0-19-853248-2 | pages=306,310,315 }}
 
* {{cite book | author=C.W. Norman | title=Undergraduate Algebra: A first course | publisher=[[Oxford University Press]] | year=1986 | isbn=0-19-853248-2 | pages=306,310,315 }}

Latest revision as of 11:46, 23 November 2009

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In mathematics, a cofactor is a component of a matrix computation of the matrix determinant.

Let M be a square matrix of size n. The (i,j) minor refers to the determinant of the (n-1)×(n-1) submatrix Mi,j formed by deleting the i-th row and j-th column from M (or sometimes just to the submatrix Mi,j itself). The corresponding cofactor is the signed determinant

The adjugate matrix adj M is the square matrix whose (i,j) entry is the (j,i) cofactor. We have

which encodes the rule for expansion of the determinant of M by any the cofactors of any row or column. This expression shows that if det M is invertible, then M is invertible and the matrix inverse is determined as

Example

Consider the following example matrix,

Its minors are the determinants (bars indicate a determinant):

The adjugate matrix of M is

and the inverse matrix is

Indeed,

and the other matrix elements of the product follow likewise.

References