A mathematical object that is useful in the analysis and solution of systems of linear equations. The determinant of a square matrix, which may represent a set of linear equations, is a number that reveals important properties of the matrix. For example, if the matrix represents a two-dimensional transformation, its determinant gives the ratio by which areas have been changed.
A determinant is a square array of numbers, each a member of a field F, used to represent another number in F. Consider the four numbers a1, a2, b1, a2. The determinant:
is described as a determinant of order 2 over F and defined to have a value
+a1b2 - a2b1.Notice that transposition (i.e., exchange of rows for columns) does not affect the value of |A|:
(since multiplication in F is commutative. This is true also of higher order determinants, so that every theorem applied to the columns of a determinant may equally be applied to its rows and vice versa. Notice also that each of the terms in +a1b2 - a2b1 contains exactly one number from each row and one from each column; thus the value of a determinant of order 2 has 2 terms, one of order 3 has 6 terms, and one of order n has n! terms (see factorial.
Other properties of determinants include:
a1x + b1y + c1 = 0,These imply that
a2(a1x + b1y + c1) - a1(a2x + b2y + c2) = 0,These may be restated, respectively, as
(a1b2 - a2b1)y - (c1a2 - c2a1) = 0The solutions, assuming that
may therefore be expressed in the form
Related category ALGEBRA
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