An invertible matrix is a square matrix 
having a matrix inverse 
, so 
, where 
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 
(0,1)-matrices:
The following table gives the numbers of invertible 
matrices over the rational
numbers 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, ... |
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."Referenced
on Wolfram|Alpha
Invertible Matrix
Cite this as:
Weisstein, Eric W. "Invertible Matrix."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InvertibleMatrix.html
Subject classifications
![[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/InvertibleMatrix/NumberedEquation1.svg)
