Cofactor

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