An important 1968 paper illustrated how an integer-preserving form of this procedure may be implemented using the Sylvester identity.  
An important 1968 paper illustrated how an integer-preserving form of this procedure may be implemented using the Sylvester identity.  This procedure has the unfortunate property of instability unless pivoting is included in its operations.  Originally mentioned by Liu Hui in The Nine Chapters on the Mathematical Art, it may be used to invert a matrix by augmenting with the identity, then reducing the unaugmented portion to the identity.  It may generally be used to solve a matrix equation by appending augmenting the coefficient matrix by the inhomogeneous vector.  FTP, name this algorithm for solving systems of linear equations named for one or two German mathematicians.
An important 1968 paper illustrated how an integer-preserving form of this procedure may be implemented using the Sylvester identity.  
Gauss-Jordan elimination (or: Gauss-Jordan method or Gaussian elimination or Gaussian method)