Matrix Inverse
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
![A=[a b; c d],](https://mathworld.wolfram.com/images/equations/MatrixInverse/NumberedEquation2.svg){kind=small} |
(2) |
the matrix inverse is
For a 
matrix
![A=[a_(11) a_(12) a_(13); a_(21) a_(22) a_(23); a_(31) a_(32) a_(33)],](https://mathworld.wolfram.com/images/equations/MatrixInverse/NumberedEquation3.svg) |
(5) |
the matrix inverse is
![A^(-1)=1/(|A|)[|a_(22) a_(23); a_(32) a_(33)| |a_(13) a_(12); a_(33) a_(32)| |a_(12) a_(13); a_(22) a_(23)|; ; |a_(23) a_(21); a_(33) a_(31)| |a_(11) a_(13); a_(31) a_(33)| |a_(13) a_(11); a_(23) a_(21)|; ; |a_(21) a_(22); a_(31) a_(32)| |a_(12) a_(11); a_(32) a_(31)| |a_(11) a_(12); a_(21) a_(22)|].](https://mathworld.wolfram.com/images/equations/MatrixInverse/NumberedEquation4.svg) |
(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
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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
Cite this as:
Stover, Christopher and Weisstein, Eric W. "Matrix Inverse." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MatrixInverse.html