a polynomial expression that is inherent in the entries of a square matrix. The size n of the square matrix, as determined from the number of entries in any row or column, is called the order of the determinant. If the entry in row i and column j is denoted as aij, then, for n=2, the determinant is a11a22-a12a21. For example, the second-order determinant
has the value (1×4)-(2×3)=4-6=-2. Its absolute value is the area of the parallelogram spanned by its row vectors. Third order determinants are similarly related to volumes. A determinant of order n is indicated by |aij|, where i and j each take on the values 1, 2, 3, … n. Its non-vanishing detects invertibility of the matrix. Its value is the sum of all terms S(π)a1π(1) … anπ(n), where π ranges over all permutations of (1, 2, … n) and S(π)=±1 is a sign called the signature of π. This value may be found more easily by expanding the determinant by minors. The minor Aij of an element aij of an nth-order determinant is the determinant of order (n-1) formed by deleting the ith row and the jth column of the original determinant. For example, in the determinant
the element a21, whose value is 3, has the minor
In expanding a determinant by minors, first the minor of every element in a particular row or column is formed. Products are derived by multiplying each minor by its corresponding element. A plus sign is placed in front of each product if the sum of the row number and column number of its element is even, and a minus sign if the sum is odd. Finally, the signed products are added algebraically. For example, expanding the above determinant by its second row yields:
Determinants of higher order can be evaluated by successive expansions of this type. By choosing rows of columns containing zeros, some terms can be eliminated. There are various rules for transforming a given determinant, which can be used to obtain a row or column most of whose elements are zeros. Determinants have many applications in mathematics and other fields, e.g., in the solution of simultaneous linear equations.