# Is a matrix similar to its RREF?

Let a matrix be denoted by A and its RREF be denoted by R. Then, is it true that R is similar to A? I am trying to find out Jordan canonical form of a large matrix. If I can somehow prove that a matrix and its RREF are similar ( i.e. they have same characteristic polynomial), it will greatly reduce my work.

It is not. Otherwise, every square invertible matrix would be similar to $I$.