# Why do elementary row operations transfer to this matrix if you multiply the original matrix on the left side with an invertible matrix?

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?

## 1 Answer

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 $$E_N \dots E_2 E_1$$?