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Transcript
Determinant
In algebra, the determinant is a special number associated to any square matrix, that is to
say, a rectangular array of numbers where the (finite) number of rows and columns are
equal. The fundamental geometric meaning of a determinant is a scale factor for measure
when the matrix is regarded as a linear transformation. Thus a 2 × 2 matrix with
determinant 2 when applied to a set of points with finite area will transform those points
into a set with twice the area. Determinants are important both in calculus, where they
enter the substitution rule for several variables, and in multilinear algebra. A matrix is
invertible if and only if its determinant is non-zero.
The determinant of a matrix A, is denoted det(A) or |A|. It is also common to denote the
determinant with elongated vertical bars:
The determinant of a matrix
is
For a fixed nonnegative integer n, there is a unique determinant function for the n×n
matrices over any commutative ring R. In particular, this unique function exists when R is
the field of real or complex numbers.
The area of the parallelogram is the absolute value of the determinant of the matrix
formed by the vectors representing the parallelogram's sides.
The 2×2 matrix
has determinant
det A = ad − bc.
The determinant det A can be viewed as the oriented area of the parallelogram with
vertices at (0,0), (a,b), (a + c, b + d), and (c,d). The oriented area is the same as the usual
area, except that it is negative when the vertices are listed in clockwise order.
Further, the parallelogram itself can be viewed as the unit square transformed by the
matrix A. The assumption here is that a linear transformation is applied to row vectors as
the vector-matrix product xTAT, where x is a column vector. The parallelogram in the
figure is obtained by multiplying matrix A (which stores the co-ordinates of our
parallelogram) with each of the row vectors
and
in
turn. These row vectors define the vertices of the unit square. With the more common
matrix-vector product Ax, the parallelogram has vertices at
and
(note that Ax = (xTAT)T ).
Thus when the determinant is equal to one, then the matrix represents an equi-areal
mapping.
The determinant of a matrix of arbitrary size can be defined by the Leibniz formula (as
explained in the next paragraph) or the Laplace formula (as explained at the end of the
Properties section).
The Leibniz formula for the determinant of an n-by-n matrix A is
Here the sum is computed over all permutations σ of the numbers {1, 2, ..., n}. A
permutation is a function that reorders this set of integers. For example, for n = 3, the
original sequence 1, 2, 3 might be reordered to 2, 3, 1 or 3, 2, 1. It is a basic fact of
combinatorics that there are n! = 1 · 2 · 3 · ... · n (n factorial) such permutations. The set
of all such permutations is denoted Sn. To any such permutation σ one attaches the
signature σ, it is +1 for even and −1 for odd permutations. Evenness or oddness can be
defined as follows: the permutation is even (odd) if the new sequence can be obtained by
an even number (odd, respectively) of switches of adjacent numbers. For example,
starting from 1, 2, 3 and switching once one gets 1, 3, 2, switching once more yields 3, 1,
2, and finally, after a total of three (an odd number) switches, one gets 3, 2, 1. Therefore
this permutation is odd. The permutation 2, 3, 1 is even (1, 2, 3 → 2, 1, 3 → 2, 3, 1, two
switches).
In any of the n factorial summands, the term
is a shorthand for the product over the indicated matrix entries, where i ranges from 1 to
n, or equivalently:
For example, for n = 4 and σ = (1, 4, 3, 2), sgn = -1 (one pair switch), and the matrix
entries are A11; A24; A33; A42. For small matrices, one gets back the formulae given in the
previous sections.
The formal extension to arbitrary dimensions was made by Tullio Levi-Civita, see (LeviCivita symbol) using a pseudo-tensor symbol. An alternative, but equivalent definition of
the determinant can be obtained by using the following theorem:
Let Mn(K) denote the set of all
matrices over the field K. There exists
exactly one function
with the two properties:
 F is alternating multilinear with regard to columns;
 F(I) = 1.
One can then define the determinant as the unique function with the above properties.