Higher-order determinants are natural generalizations. The minor of the entry in the th-order determinant is the ()th-order determinant derived from by deleting the th row and the th column. The cofactor of is
An th-order determinant expanded by its th row is given by
If two rows (or columns) of a determinant are interchanged, then the determinant changes sign. If two rows (columns) of a determinant are identical, then the determinant is zero. If all the elements of a row (column) of a determinant are multiplied by an arbitrary factor , then the result is a determinant which is times the original. If times a row (column) of a determinant is added to another row (column), then the value of the determinant is unchanged.
For real-valued ,
Compare also (1.3.7) for the left-hand side. Equality holds iff
for every distinct pair of , or when one of the factors vanishes.
An alternant is a determinant function of variables which changes sign when two of the variables are interchanged. Examples:
where are the th roots of unity (1.11.21).
Let be defined for all integer values of and , and denote the determinant
If tends to a limit as , then we say that the infinite determinant converges and .
Of importance for special functions are infinite determinants of Hill’s type. These have the property that the double series
converges (§1.9(vii)). Here is the Kronecker delta. Hill-type determinants always converge.