A unimodular matrix is a real square matrix
with determinant (Born and Wolf 1980, p. 55; Goldstein 1980,
p. 149). More generally, a matrix with elements in the polynomial domain of a field is called unimodular if it has an inverse whose elements are
also in .
A matrix
is therefore unimodular iff its determinant
is a unit of
(MacDuffee 1943, p. 137).

The matrix inverse of a unimodular real
matrix is another unimodular matrix.

There are an infinite number of unimodular matrices not containing any 0s or . One parametric family is

(1)

Specific examples of unimodular matrices having small positive integer entries include

(2)

(Guy 1989, 1994).

The th
power of a unimodular matrix

(3)

is given by

(4)

where

(5)

and the
are Chebyshev polynomials of the
second kind ,

(6)

(Born and Wolf 1980, p. 67).

See also Chebyshev Polynomial of the Second Kind ,

Determinant ,

Identity
Matrix ,

Unit Matrix
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References Born, M. and Wolf, E. Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of
Light, 6th ed. New York: Pergamon Press, pp. 55 and 67, 1980. Goldstein,
H. Classical
Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 149, 1980. 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. MacDuffee, C. C. Vectors
and Matrices. Washington, DC: Math. Assoc. Amer., 1943. Séroul,
R. Programming
for Mathematicians. Berlin: Springer-Verlag, p. 162, 2000. Referenced
on Wolfram|Alpha Unimodular Matrix
Cite this as:
Weisstein, Eric W. "Unimodular Matrix."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/UnimodularMatrix.html

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