# Cofactor

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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=44605
This article was adapted from an original article by V.N. Remeslennikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article