To add a formal answer to the question, changing the position of the variables in the regressor matrix $$ X=(X_1,\ldots,X_p), $$ where $X_j=(x_{1j},\ldots,x_{nj})'$, amounts to postmultiplying $X$ with a $(p\times p)$ [permutation matrix $P$](https://en.wikipedia.org/wiki/Permutation_matrix) that has a single entry 1 in each column $j$ that indicates the new column position of that regressor $X_j$. For example, if the new columns are to be the old columns 2, 1 and 3, we have $$P=\begin{pmatrix} 0&1&0\\ 1&0&0\\ 0&0&1 \end{pmatrix}$$ This matrix $P$ is invertible, being just a permuted version of the identity matrix. Also, as is easy to check that $P'P=I$, $P^{-1}$ is equal to the transpose of $P$, $P^{-1}=P'$. Thus, the OLS coefficient of the regression of $y$ on the transformed regressors is \begin{eqnarray*} \hat\beta_t&=&((XP)'XP)^{-1}(XP)'y\\ &\stackrel{(AB)'=B'A'}{=}&(P'X'XP)^{-1}P'X'y\\ &\stackrel{(ABC)^{-1}=C^{-1}B^{-1}A^{-1}}{=}&P^{-1}(X'X)^{-1}\underbrace{(P')^{-1}P'}_{=I}X'y\\ &=&P^{-1}(X'X)^{-1}X'y\\ &=&P'(X'X)^{-1}X'y\\ &=&P'\hat\beta\\ \end{eqnarray*} Here, $P'$ is a matrix that permutes the *row* elements of $\hat\beta$, and hence permutes the entries of the coeffient estimator according to the permutation of the columns.