How prove that the elementary operations don't change the rank of a matrix One considers certain operations, called elementary row operations, that are applied to a matrix $A$ to obtain a new matrix $B$ of the same size.
These are the following:

*

*exchange rows $i_1$ and $i_2$ of $A$ (where $i_1\neq i_2$);

*replace row $i_1$ of $A$ by itself plus the scalar $c$ times row $i_2$ (where $i_1\neq i_2$);

*multiply row $i$ of $A$ by the non-zero scalar $\lambda$.

Naturally this operations can be implemented on a column and so we would call the analogous operations on the columns "elementary column operations".
Theorem
If $B$ is the matrix obtained by applying an elementary row/column operation to $A$, then these two matrix has the same rank.
Unfortunately I'm not able to prove the previous theorem, so could someone help me, please?

This is a super important linear algebra theorem. The basic idea of the proof is that each of these operations is equivalent to right-multiplication by a matrix of full rank. I'll give an example of each operation in the 2 by 2 case:

*

*Swap the rows by multiplying on the right by
\begin{pmatrix}
0 & 1 \\
1 & 0 
\end{pmatrix}

*Add the top row to the bottom with
\begin{pmatrix}
1 & 1 \\
0 & 1 
\end{pmatrix}

*Scale the top row by $c$ using
\begin{pmatrix}
c & 0 \\
0 & 1 
\end{pmatrix}
Since multiplying by a matrix of full rank preserves rank, it follows that the elementary row operations are rank-preserving.
