# Difference between revisions of "Cofactor"

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(See also Adjugate matrix) |
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− | ''for a [[ | + | ''for a [[minor]] $M$'' |

The number | 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. | |

− | |||

− | where | ||

====Comments==== | ====Comments==== | ||

− | This Laplace theorem is often referred to as Laplace's development of a determinant. | + | This Laplace theorem is often referred to as Laplace's development of a determinant. See also [[Adjugate matrix]]. |

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.W. Turnball, "The theory of determinants, matrices, and invariants" , Dover, reprint ( | + | <table> |

+ | <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> | ||

+ | </table> | ||

+ | |||

+ | {{TEX|done}} |

## Latest revision as of 11:43, 9 February 2021

*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. See also Adjugate matrix.

#### 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