The inverse of a square matrix , sometimes called a reciprocal matrix, is a matrix such that
(1)
where
is the identity matrix . Courant and Hilbert (1989,
p. 10) use the notation to denote the inverse matrix.
A square matrix has an inverse iff the determinant (Lipschutz 1991, p. 45). The
so-called invertible matrix theorem is
major result in linear algebra which associates the existence of a matrix inverse
with a number of other equivalent properties. A matrix possessing an inverse is called
nonsingular , or invertible.
The matrix inverse of a square matrix may be taken in the Wolfram
Language using the function Inverse [m ].
For a matrix
(2)
the matrix inverse is
For a matrix
(5)
the matrix inverse is
(6)
A general
matrix can be inverted using methods such as the Gauss-Jordan
elimination , Gaussian elimination , or
LU decomposition .
The inverse of a product of matrices and can be expressed in terms of and . Let
(7)
Then
(8)
and
(9)
Therefore,
(10)
so
(11)
where
is the identity matrix , and
(12)
See also Drazin Inverse ,
Gauss-Jordan Elimination ,
Gaussian Elimination ,
LU Decomposition ,
Matrix ,
Matrix 1-Inverse ,
Matrix
Addition ,
Matrix Multiplication ,
Moore-Penrose Matrix Inverse ,
Nonsingular
Matrix ,
Pseudoinverse ,
Singular
Matrix ,
Strassen Formulas Explore this topic in the MathWorld classroom
Portions of this entry contributed by Christopher
Stover
Explore with Wolfram|Alpha
References Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, p. 11,
1962. Ben-Israel, A. and Greville, T. N. E. Generalized
Inverses: Theory and Applications. New York: Wiley, 1977. Courant,
R. and Hilbert, D. Methods
of Mathematical Physics, Vol. 1. New York: Wiley, 1989. Jodár,
L.; Law, A. G.; Rezazadeh, A.; Watson, J. H.; and Wu, G. "Computations
for the Moore-Penrose and Other Generalized Inverses." Congress. Numer. 80 ,
57-64, 1991. Lipschutz, S. "Invertible Matrices." Schaum's
Outline of Theory and Problems of Linear Algebra, 2nd ed. New York: McGraw-Hill,
pp. 44-45, 1991. Nash, J. C. Compact
Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed.
Bristol, England: Adam Hilger, pp. 24-26, 1990. Press, W. H.;
Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Is
Matrix Inversion an
Process?" §2.11 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 95-98, 1992. Rosser, J. B.
"A Method of Computing Exact Inverses of Matrices with Integer Coefficients."
J. Res. Nat. Bur. Standards Sect. B. 49 , 349-358, 1952. Referenced
on Wolfram|Alpha Matrix Inverse
Cite this as:
Stover, Christopher and Weisstein, Eric W. "Matrix Inverse." From MathWorld --A
Wolfram Web Resource. https://mathworld.wolfram.com/MatrixInverse.html
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