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Invertible Matrix

An invertible matrix is a square matrix A having a matrix inverse A^(-1), so AA^(-1)=A^(-1)A=I, where I is the identity matrix. Invertible matrices are also called nonsingular or regular matrices. Over a field, invertibility is equivalent to having nonzero determinant (Lipschutz 1991, p. 45). For example, over the rational numbers, there are 6 invertible 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 invertible n×n matrices over the rational numbers 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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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."

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Invertible Matrix

Cite this as:

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

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