# 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.