Nonsingular Matrix

A square matrix that is not singular, i.e., one that has a matrix inverse. Nonsingular matrices are sometimes also called regular matrices. A square matrix is nonsingular iff its determinant is nonzero (Lipschutz 1991, p. 45). For example, there are 6 nonsingular 2×2 (0,1)-matrices:

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

The following table gives the numbers of nonsingular n×n matrices for certain matrix classes.

matrix typeOEIScounts for n=1, 2, ...
(-1,0,1)-matricesA0569892, 48, 11808, ...
(-1,1)-matricesA0569902, 8, 192, 22272, ...
(0,1)-matricesA0551651, 6, 174, 22560, ...

See also

Determinant, Diagonalizable Matrix, Invertible Matrix Theorem, Matrix Inverse, Singular Matrix

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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.Lipschutz, S. "Invertible Matrices." Schaum's Outline of Theory and Problems of Linear Algebra, 2nd ed. New York: McGraw-Hill, pp. 44-45, 1991.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 A055165, A056989, and A056990 in "The On-Line Encyclopedia of Integer Sequences."

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Nonsingular Matrix

Cite this as:

Weisstein, Eric W. "Nonsingular Matrix." From MathWorld--A Wolfram Web Resource.

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