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# 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 (0,1)-matrices:

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

 matrix type OEIS counts for , 2, ... -matrices A056989 2, 48, 11808, ... -matrices A056990 2, 8, 192, 22272, ... -matrices A055165 1, 6, 174, 22560, ...

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

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## References

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."

## Referenced on Wolfram|Alpha

Nonsingular Matrix

## Cite this as:

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