Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations.
As shown by Cramer's rule, a nonhomogeneous
system of linear equations has a unique solution if
the determinant of the system's matrix
is nonzero (i.e., the matrix is nonsingular). For example, eliminating ,
, and
from the equations
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(1)
|
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(2)
|
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(3)
|
gives the expression
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(4)
|
which is called the determinant for this system of equation. Determinants are defined only for square matrices.
If the determinant of a matrix is 0, the matrix is said to be singular, and if the determinant is 1, the matrix is said to be unimodular.
The determinant of a matrix ,
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(5)
|
is commonly denoted ,
, or in component
notation as
,
,
or
(Muir). Note that
the notation
may be more convenient when indicating
the absolute value of a determinant, i.e.,
instead
of
. The determinant is implemented in Mathematica as Det[m].
A determinant is defined to be
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(6)
|
A determinant can be expanded "by minors" to obtain
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(7)
|
A general determinant for a matrix has a value
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(8)
|
with no implied summation over and where
(also denoted
) is the cofactor of
defined by
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(9)
|
and is the minor
of matrix
formed by eliminating
row
and column
from
. This process is
called determinant
expansion by minors (or "Laplacian expansion by minors," sometimes
further shortened to simply "Laplacian expansion").
A determinant can also be computed by writing down all permutations of , taking
each permutation as the subscripts of the letters
,
, ..., and summing
with signs determined by
, where
is the number
of permutation inversions
in permutation
(Muir 1960, p. 16), and
is the permutation symbol.
For example, with , the permutations and the number of
inversions they contain are 123 (0), 132 (1), 213 (1), 231 (2), 312 (2), and 321
(3), so the determinant is given by
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(10)
|
If is a constant and
an
square matrix, then
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(11)
|
Given an determinant, the additive inverse
is
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(12)
|
Determinants are also distributive, so
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(13)
|
This means that the determinant of a matrix inverse can be found as follows:
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(14)
|
where is the identity matrix, so
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(15)
|
Determinants are multilinear in rows and columns, since
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(16)
|
and
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(17)
|
The determinant of the similarity transformation of a matrix is equal to the determinant of the original matrix
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(18)
|
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(19)
|
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(20)
|
The determinant of a similarity transformation minus a multiple of the unit matrix is given by
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(21)
|
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(22)
|
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(23)
|
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(24)
|
The determinant of a transpose equals the determinant of the original matrix,
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(25)
|
and the determinant of a complex conjugate is equal to the complex conjugate of the determinant
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(26)
|
Let be a small number. Then
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(27)
|
where is the matrix trace of
. The determinant
takes on a particularly simple form for a triangular
matrix
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(28)
|
Important properties of the determinant include the following, which include invariance under elementary row and column operations.
1. Switching two rows or columns changes the sign.
2. Scalars can be factored out from rows and columns.
3. Multiples of rows and columns can be added together without changing the determinant's value.
4. Scalar multiplication of a row by a constant multiplies the
determinant by
.
5. A determinant with a row or column of zeros has value 0.
6. Any determinant with two rows or columns equal has value 0.
Property 1 can be established by induction. For a matrix, the determinant is
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(29)
|
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(30)
|
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(31)
|
For a matrix,
the determinant is
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(32)
|
Property 2 follows likewise. For and
matrices,
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(33)
|
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(34)
|
and
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(35)
|
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(36)
|
Property 3 follows from the identity
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(37)
|
If is an
matrix with
real numbers, then
has
the interpretation as the oriented
-dimensional content of the parallelepiped spanned by the column vectors
, ...,
in
. Here, "oriented"
means that, up to a change of
or
sign, the number is the
-dimensional content, but the sign depends on the "orientation" of the column vectors
involved. If they agree with the standard orientation, there is a
sign; if not, there is a
sign. The parallelepiped
spanned by the
-dimensional vectors
through
is the collection of points
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(38)
|
where is a real
number in the closed interval
.
Hadamard showed that the absolute value of the determinant of a complex matrix
with entries in the unit disk satisfies
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(39)
|
(Brenner). The plots above show the distribution of determinants for random complex matrices with entries satisfying
for
, 3, and 4.