## Matrix Inverse

A Matrix has an inverse Iff the Determinant . For a Matrix

 (1)

the inverse is
 (2)

For a Matrix,
 (3)

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

 (4)

Then
 (5)

and
 (6)

Therefore,
 (7)

so
 (8)

where I is the Identity Matrix, and
 (9)

References

Ben-Israel, A. and Greville, T. N. E. Generalized Inverses: Theory and Applications. New York: Wiley, 1977.

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.