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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 | Sloane | counts for  , 2, ... |  -matrices | A056989 | 2, 48, 11808, ... |  -matrices | A056990 | 2, 8, 192, 22272, ... |  -matrices | A055165 | 1, 6, 174, 22560, ... |
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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![[0 1; 1 0],[0 1; 1 1],[1 0; 0 1],[1 0; 1 1],[1 1; 0 1],[1 1; 1 0].](/images/equations/NonsingularMatrix/NumberedEquation1.gif)
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