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Show that a matrix $ X \in M_{m,n}(K)$ is exactly than transferable with elementary row operations into a matrix $ X' \in M_{m,n}(K)$ , if $X'=TX$, if T is an invertible matrix.

It's clear, that we have 3 elementary row operations.

  • row switching
  • row multiplication
  • row addition

I assume, that these row operations are an equivalence relation. Can I maybe use here somehow that these row operations are making an equivalence relation?

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Hint: you should show that performing any of the three elementary row operations on a matrix $X$ to get $Y$ is equivalent to multiplying $X$ on the left by some invertible matrix $E$: $Y = EX$. Note that the matrix $E$ will be different depending on which operation is performed.

Then performing a sequence of row operations is equivalent to left multiplying $X$ by a sequence of invertible matrices $E_1, E_2, \dots, E_N$, so $X' = E_N \dots E_2 E_1 X$. What can you say about the invertibility of $E_N \dots E_2 E_1$?

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