What is Routh Hurwitz Stability Criterion?

As the name suggests, this criterion is given by two mathematician Adolf Hurwitz and Edward John Routh independently. It is a necessary test to validate the stability of a linear time-variant control system. The stability of the system is possible only

1. if either the roots of the first column have the same sign
2. if any sign is changed then the number of sign changes in the first column should be equal to the number of roots with positive real parts.

Thus, it shows the path to determine if equations of the linear system have stable solutions without directly solving the system.