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''for a [[Minor|minor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c0229501.png" />''
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''for a [[minor]] $M$''
  
 
The number
 
The number
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$$
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(-1)^{s+t} \det A_{i_1\ldots i_k}^{j_1\ldots j_k}
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c0229502.png" /></td> </tr></table>
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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.
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c0229503.png" /> is a minor of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c0229504.png" />, with rows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c0229505.png" /> and columns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c0229506.png" />, of some square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c0229507.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c0229508.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c0229509.png" /> is the determinant of the matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c02295010.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c02295011.png" /> by deletion of the rows and columns of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c02295012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c02295013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c02295014.png" />. Laplace's theorem is valid: If any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c02295015.png" /> rows are fixed in a determinant of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c02295016.png" />, then the sum of the products of the minors of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022950/c02295017.png" />-th order corresponding to the fixed rows by their cofactor is equal to the value of this determinant.
 
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.W. Turnball,  "The theory of determinants, matrices, and invariants" , Dover, reprint  (1980)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.W. Turnball,  "The theory of determinants, matrices, and invariants" , Dover, reprint  (1960) {{ZBL|0103.00702}}</TD></TR>
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</table>
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 +
{{TEX|done}}

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=15198
This article was adapted from an original article by V.N. Remeslennikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article