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?