Theorem 3: Let $E$ be the elementary matrix required to pre-multiply a square matrix $A$ for making an elementary row operation on $A$. Then, $E^T$ is the elementary matrix post-multiplying $A$ for making the corresponding elementary column operation.
I am aware of two theorems and their proofs:
Theorem 1 : Making an elementary row operation ($R_{ij},R_i(\alpha),R_{ij}(\beta)$) on a matrix $A$ is equivalent to pre-multiplying $A$ by the corresponding elementary matrix ($E_{ij},E_i(\alpha),E_{ij}(\beta)$ respectively).
Theorem 2 : Making the column operations $C_{ij},C_i(\alpha)$ and $C_{ij}(\beta)$ on a matrix $A$ is equivalent to post-multiplying $A$ by the elementary matrices $E_{ij},E_i(\alpha)$ and $E_{ji}(\beta)$ respectively.
Question : Is using theorems 1 and 2 together as a very basic/loose(?) proof for theorem 3 wrong? If there is a formal proof for theorem 3, please provide it.
EDIT
Notations:
$R_{ij}/C_{ij}$ – interchanging $i^{th}$ and $j^{th}$ rows/columns
$R_i(\alpha)/C_i(\alpha)$ – multiplying $i^{th}$ row/column by $\alpha$
$R_{ij}(\beta)/C_{ij}(\beta)$ – adding $\beta$ times $j^{th}$ row to $i^{th}$ row/column
$E_{ij},E_i(\alpha),E_{ij}(\beta)$ are the matrices obtained by performing the corresponding row operations on $I$.
Best Answer
I do not understand the notation you are using for elementary row operations and such, but theorem 3 can be proved immediately from the fact that $(AB)^T=B^TA^T$.
Indeed, let $E$ be an elementary row operation. Then $AE^T = (EA^T)^T$.
Since $E$ is an elementary row operation, we will be operating over a row of $A^T$. But an operation over a row of $A^T$ coincides with the same column operation over $A$.