# Difference between revisions of "Cofactor"

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The number | The number | ||

$$ | $$ | ||

− | (-1)^{s+t} \det A_{i_1\ | + | (-1)^{s+t} \det A_{i_1\cdots i_k}^{j_1\cdots j_k} |

$$ | $$ | ||

− | where $M$ is a minor of order $k$, with rows $i_1,\ | + | 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. |

## Revision as of 12:57, 14 February 2020

*for a minor $M$*

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

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.

#### 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=39509