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Singular Matrix

A singular matrix over a field is a square matrix that does not have a matrix inverse. A matrix is singular iff its determinant is 0. For example, there are 10 singular 2×2 (0,1)-matrices:

[0 0; 0 0],[0 0; 0 1],[0 0; 1 0],[0 0; 1 1],[0 1; 0 0] [0 1; 0 1],[1 0; 0 0],[1 0; 1 0],[1 1; 0 0],[1 1; 1 1].

The following table gives the numbers of singular n×n matrices for certain matrix classes, with singularity taken over the rational numbers Q and row and column positions distinguished.

matrix typeOEIScounts for n=1, 2, ...
(-1,0,1)-matricesA0579811, 33, 7875, 15099201, ...
(-1,1)-matricesA0579820, 8, 320, 43264, ...
(0,1)-matricesA0467471, 10, 338, 42976, ...

See also

Determinant, Ill-Conditioned Matrix, Matrix Inverse, Nonsingular Matrix, Singular Value Decomposition

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References

Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, p. 39, 1962.Faddeeva, V. N. Computational Methods of Linear Algebra. New York: Dover, p. 11, 1958.Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins, p. 51, 1996.Kahn, J.; Komlós, J.; and Szemerédi, E. "On the Probability That a Random +/-1-Matrix Is Singular." J. Amer. Math. Soc. 8, 223-240, 1995. https://doi.org/10.1090/S0894-0347-1995-1260107-2.Komlós, J. "On the Determinant of (0,1)-Matrices." Studia Sci. Math. Hung. 2, 7-21, 1967.Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, p. 70, 1988.Marcus, M. and Minc, H. A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, p. 3, 1992.Sloane, N. J. A. Sequences A046747, A057981, and A057982 in "The On-Line Encyclopedia of Integer Sequences."

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Singular Matrix

Cite this as:

Weisstein, Eric W. "Singular Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SingularMatrix.html

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