Determinants are mathematical objects which are very useful in the analysis and solution of systems of linear equations. As
shown in Cramer's Rule, a nonhomogeneous system of linear equations has a nontrivial solution Iff the
determinant of the system's Matrix is Nonzero (so that the Matrix is nonsingular). A
determinant is defined to be
![\begin{displaymath}
\mathop{\rm det}\nolimits \left[{\matrix{a & b\cr c & d\cr}}...
... \left\vert\matrix{a & b\cr c & d\cr}\right\vert \equiv ad-bc.
\end{displaymath}](d1_645.gif) |
(1) |
A
determinant can be expanded by Minors to obtain
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(2) |
A general determinant for a Matrix A has a value
![\begin{displaymath}
\vert{\hbox{\sf A}}\vert = \sum_i a_{ij}a^{ij},
\end{displaymath}](d1_649.gif) |
(3) |
with no implied summation over
and where
is the Cofactor of
defined by
![\begin{displaymath}
a^{ij} \equiv (-1)^{i+j} C_{ij}.
\end{displaymath}](d1_653.gif) |
(4) |
Here, C is the
Matrix formed by eliminating row
and column
from A, i.e., by
Determinant Expansion by Minors.
Given an
determinant, the additive inverse is
![\begin{displaymath}
\vert-{\hbox{\sf A}}\vert = (-1)^n\vert{\hbox{\sf A}}\vert.
\end{displaymath}](d1_656.gif) |
(5) |
Determinants are also Distributive, so
![\begin{displaymath}
\vert{\hbox{\sf A}}{\hbox{\sf B}}\vert = \vert{\hbox{\sf A}}\vert\,\vert{\hbox{\sf B}}\vert.
\end{displaymath}](d1_657.gif) |
(6) |
This means that the determinant of a Matrix Inverse can be found as follows:
![\begin{displaymath}
\vert{\hbox{\sf I}}\vert=\vert{\hbox{\sf A}}{\hbox{\sf A}}^{...
...= \vert{\hbox{\sf A}}\vert\,\vert{\hbox{\sf A}}^{-1}\vert = 1,
\end{displaymath}](d1_658.gif) |
(7) |
where I is the Identity Matrix, so
![\begin{displaymath}
\vert{\hbox{\sf A}}\vert={1\over \vert{\hbox{\sf A}}^{-1}\vert}.
\end{displaymath}](d1_659.gif) |
(8) |
Determinants are Multilinear in rows and columns, since
![\begin{displaymath}
\left\vert\matrix{a_1 & a_2 & a_3\cr a_4 & a_5 & a_6\cr a_7 ...
...& 0 & a_3\cr a_4 & a_5 & a_6\cr a_7 & a_8 & a_9\cr}\right\vert
\end{displaymath}](d1_660.gif) |
(9) |
and
![\begin{displaymath}
\left\vert\matrix{a_1 & a_2 & a_3\cr a_4 & a_5 & a_6\cr a_7 ...
...0 & a_2 & a_3\cr 0 & a_5 & a_6\cr a_7 & a_8 & a_9}\right\vert.
\end{displaymath}](d1_661.gif) |
(10) |
The determinant of the Similarity Transformation of a matrix is equal to the determinant of the original
Matrix
![\begin{displaymath}
\vert{\hbox{\sf B}}{\hbox{\sf A}}{\hbox{\sf B}}^{-1}\vert = ...
... {1\over \vert{\hbox{\sf B}}\vert} = \vert{\hbox{\sf A}}\vert.
\end{displaymath}](d1_662.gif) |
(11) |
The determinant of a similarity transformation minus a multiple of the unit Matrix is given by
The determinant of a Matrix Transpose equals the determinant of the original Matrix,
![\begin{displaymath}
\vert{\hbox{\sf A}}\vert = \vert{\hbox{\sf A}}^{\rm T}\vert,
\end{displaymath}](d1_666.gif) |
(13) |
and the determinant of a Complex Conjugate is equal to the Complex Conjugate of the determinant
![\begin{displaymath}
\vert{\hbox{\sf A}}^*\vert = \vert{\hbox{\sf A}}\vert^*.
\end{displaymath}](d1_667.gif) |
(14) |
Let
be a small number. Then
![\begin{displaymath}
\vert{\hbox{\sf I}}+\epsilon{\hbox{\sf A}}\vert = 1+\epsilon...
...op{\rm Tr}\nolimits ({\hbox{\sf A}})+{\mathcal O}(\epsilon^2),
\end{displaymath}](d1_668.gif) |
(15) |
where
is the Trace of A. The determinant takes on a particularly simple form for a
Triangular Matrix
![\begin{displaymath}
\left\vert\matrix{
a_{11} & a_{21} & \cdots & a_{k1}\cr
0 ...
... & 0 & \cdots & a_{kk}\cr}\right\vert
= \prod_{n=1}^k a_{nn}.
\end{displaymath}](d1_670.gif) |
(16) |
Important properties of the determinant include the following.
- 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
For a
Matrix, the determinant is
Property 2 follows likewise. For
and
matrices,
![\begin{displaymath}
\left\vert\matrix{ka_1 & b_1\cr ka_2 & b_2\cr}\right\vert = ...
..._2) = k\left\vert\matrix{a_1 & b_1\cr a_2 & b_2\cr}\right\vert
\end{displaymath}](d1_679.gif) |
(19) |
and
![\begin{displaymath}
\left\vert\matrix{ka_1 & b_1 & c_1\cr ka_2 & b_2 & c_2\cr ka...
..._1 & c_1\cr a_2 & b_2 & c_2\cr a_3 & b_3 & c_3\cr}\right\vert.
\end{displaymath}](d1_680.gif) |
(20) |
Property 3 follows from the identity
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(21) |
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
-D vectors
through
is the collection
of points
![\begin{displaymath}
t_1 {\bf v}_1 + \ldots + t_i {\bf v}_i,
\end{displaymath}](d1_691.gif) |
(22) |
where
is a Real Number in the Closed Interval [0,1].
There are an infinite number of
determinants with no 0 or
entries having unity determinant. One
parametric family is
![\begin{displaymath}
\left\vert\matrix{
-8n^2-8n & 2n+1 & 4n\cr
-4n^2-4n & n+1 & 2n+1\cr
-4n^2-4n-1 & n & 2n-1\cr}\right\vert.
\end{displaymath}](d1_693.gif) |
(23) |
Specific examples having small entries include
![\begin{displaymath}
\left\vert\matrix{2 & 3 & 2\cr 4 & 2 & 3\cr 9 & 6 & 7\cr}\ri...
...x{2 & 3 & 6\cr 3 & 2 & 3\cr 17 & 11 & 16\cr}\right\vert, \dots
\end{displaymath}](d1_694.gif) |
(24) |
(Guy 1989, 1994).
See also Circulant Determinant, Cofactor, Hessian Determinant, Hyperdeterminant, Immanant,
Jacobian, Knot Determinant, Matrix, Minor, Permanent, Vandermonde Determinant,
Wronskian
References
Arfken, G. ``Determinants.'' §4.1 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 168-176, 1985.
Guy, R. K. ``Unsolved Problems Come of Age.'' Amer. Math. Monthly 96, 903-909, 1989.
Guy, R. K. ``A Determinant of Value One.'' §F28 in
Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 265-266, 1994.
© 1996-9 Eric W. Weisstein
1999-05-24