Gauss-Jordan Elimination

A method for finding a Matrix Inverse. To apply Gauss-Jordan elimination, operate on a Matrix

$$\left[{\matrix{{\hbox{\sf A}}& {\hbox{\sf I}}\cr}}\right] \e... ...cr a_{n1} & \cdots & a_{nn} & 0 & 0 & \cdots & 1\cr}}\right],$$

where I is the Identity Matrix, to obtain a Matrix of the form

$$\left[{\matrix{ 1 & 0 & \cdots & 0 & b_{11} & \cdots & b_{1... ...cr 0 & 0 & \cdots & 1 & b_{n1} & \cdots & b_{nn}\cr}}\right].$$

The Matrix

$${\hbox{\sf B}}\equiv \left[{\matrix{ b_{11} & \cdots & b_{1... ...ots & \ddots & \vdots\cr b_{n1} & \cdots & b_{nn}\cr}}\right]$$

is then the Matrix Inverse of A. The procedure is numerically unstable unless Pivoting (exchanging rows and columns as appropriate) is used. Picking the largest available element as the pivot is usually a good choice.

See also Gaussian Elimination, LU Decomposition, Matrix Equation

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Gauss-Jordan Elimination'' and ``Gaussian Elimination with Backsubstitution.'' §2.1 and 2.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 27-32 and 33-34, 1992.