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],](/images/equations/MatrixInverse/NumberedEquation2.gif) |
(2)
|
the matrix inverse is
 |  | ![1/(|A|)[d -b; -c a]](/images/equations/MatrixInverse/Inline11.gif) |
(3)
|
 |  | ![1/(ad-bc)[d -b; -c a].](/images/equations/MatrixInverse/Inline14.gif) |
(4)
|
For a 
matrix
![A=[a_(11) a_(12) a_(13); a_(21) a_(22) a_(23); a_(31) a_(32) a_(33)],](/images/equations/MatrixInverse/NumberedEquation3.gif) |
(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)|].](/images/equations/MatrixInverse/NumberedEquation4.gif) |
(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)
|
Portions of this entry contributed by Christopher Stover
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.
Stover, Christopher and Weisstein, Eric W. "Matrix Inverse." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MatrixInverse.html
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