subdeterminant

A subdeterminant is a determinant of a submatrix. A submatrix is a matrix obtained from a larger matrix by deleting any number of rows and columns. The order of a subdeterminant is the number of rows (or columns) in the submatrix from which it is calculated. For an n x n matrix, subdeterminants can be of any order from 1 to n.

Subdeterminants play a role in various areas of linear algebra and its applications. For example, in the study of rank of a matrix, a non-zero subdeterminant of a certain order indicates that the rank of the matrix is at least that order. Conversely, if all subdeterminants of a certain order are zero, it implies the rank is less than that order.

The minors of a matrix are a specific type of subdeterminant. A minor of order k of an n x n matrix is the determinant of a k x k submatrix. The cofactor of an element in a matrix is related to a minor, specifically it is the minor multiplied by (-1) raised to the power of the sum of the row and column indices of the element. These concepts are fundamental to calculating determinants of larger matrices using cofactor expansion.