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?

The elementary operations have elementary matrices associated to them.
These matrices are invertible, thus the product of your original matrix by one of these does not change its rank, since the number of linearly independent row\column vectors is conserved by invertible linear transformations.
