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Difference between revisions of "Cofactor"

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m (dots)
m
 
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The number
 
The number
 
$$
 
$$
(-1)^{s+t} \det A_{i_1\cdots i_k}^{j_1\cdots j_k}
+
(-1)^{s+t} \det A_{i_1\ldots i_k}^{j_1\ldots j_k}
 
$$
 
$$
  
where $M$ is a minor of order $k$, with rows $i_1,\dotsc,i_k$ and columns $j_1,\dotsc,j_k$, of some square matrix $A$ of order $n$; $\det A_{i_1\cdots i_k}^{j_1\cdots j_k}$ is the determinant of the matrix of order $n-k$ obtained from $A$ by deletion of the rows and columns of $M$; $s = i_1 + \dotsb + i_k$, $t = j_1 + \dotsb + j_k$. Laplace's theorem is valid: If any $r$ rows are fixed in a determinant of order $n$, then the sum of the products of the minors of the $r$-th order corresponding to the fixed rows by their cofactor is equal to the value of this determinant.
+
where $M$ is a minor of order $k$, with rows $i_1,\ldots,i_k$ and columns $j_1,\ldots,j_k$, of some square matrix $A$ of order $n$; $\det A_{i_1\ldots i_k}^{j_1\ldots j_k}$ is the determinant of the matrix of order $n-k$ obtained from $A$ by deletion of the rows and columns of $M$; $s = i_1 + \cdots + i_k$, $t = j_1 + \cdots + j_k$. Laplace's theorem is valid: If any $r$ rows are fixed in a determinant of order $n$, then the sum of the products of the minors of the $r$-th order corresponding to the fixed rows by their cofactor is equal to the value of this determinant.
  
  

Latest revision as of 12:57, 14 February 2020

for a minor $M$

The number $$ (-1)^{s+t} \det A_{i_1\ldots i_k}^{j_1\ldots j_k} $$

where $M$ is a minor of order $k$, with rows $i_1,\ldots,i_k$ and columns $j_1,\ldots,j_k$, of some square matrix $A$ of order $n$; $\det A_{i_1\ldots i_k}^{j_1\ldots j_k}$ is the determinant of the matrix of order $n-k$ obtained from $A$ by deletion of the rows and columns of $M$; $s = i_1 + \cdots + i_k$, $t = j_1 + \cdots + j_k$. Laplace's theorem is valid: If any $r$ rows are fixed in a determinant of order $n$, then the sum of the products of the minors of the $r$-th order corresponding to the fixed rows by their cofactor is equal to the value of this determinant.


Comments

This Laplace theorem is often referred to as Laplace's development of a determinant.

References

[a1] H.W. Turnball, "The theory of determinants, matrices, and invariants" , Dover, reprint (1960) Zbl 0103.00702
How to Cite This Entry:
Cofactor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cofactor&oldid=44606
This article was adapted from an original article by V.N. Remeslennikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article