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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.

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

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