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My understanding of the definition of matrix equivalence is that two matrices are equal if they can be transformed into each other by a combination of elementary row and column operations.

Therefore, if A = PB, where P is an invertible matrix composed of elementary matrices, then A and B are equivalent matrices because they can be transformed into each other by elementary row and column operations.

As such, A and B have equivalent reduced row echelon forms, and equivalent images and kernels, because A is equivalent to B.

This seems too simplistic, and makes short work of a proof I am working on if it is true. Is something about my logic here incorrect?

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  • $\begingroup$ I can't tell what you are talking about, but you should look up Smith Normal Form. $\endgroup$ – Will Jagy Mar 10 '18 at 5:09
  • $\begingroup$ There are many different types of equivalence. If $A=PB$ as above, then $A,B$ have the same kernel, but the images may differ. $\endgroup$ – copper.hat Mar 10 '18 at 5:10

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