A brief history.

The solution to solving system problems by use of Matrices trace their roots back to the 2nd century BC and the Chinese "Fangcheng". It was only towards the end of the 17th century and primarily the work of Carl Friedrich Gauss, known as the "Prince of Mathematicians", that gave rise to what is now known as the Gaussian Methods(or modified forms named "LU Decomposition"). Until now; these Gaussian Methods were the primary algorithms of our day; recognized in modern textbooks and universities as the standard in solving system problems - that is, until today.

The Matrix or System

Cona Inverse Rotation
Today; with the discovery of the Cona Inverse Rotation, the 21st Century mathematical approach of beauty, simplicity, and modernity is borne. The natural computational flow of the algorithm manifests from a non-traditional unified dual-rotation which builds upon the mathematical ideas of Wallace Givens and the Givens Rotation. According to John J. Cona, who made the discovery, "when we think of solving a system problem similar to Ax = b, most textbooks describe and most algorithms rely on some derivation of a Gaussian Method – with this new mathematical discovery - No Archaic RREF or Gaussian Methods (such as "LU Decomposition") are required – just beautiful elegant mathematical rotations." Official US Copyright Text (Here)