<s>[INST] Least square regression weight estimation for $\beta_1$ and $\beta_2$? So, I have been looking at this post, and others that are similar and know that the least square estimation of $\beta_1,\beta_2$ will be $(X^TX)^{-1}X^TY$, where the model is $Y_i = \beta_1x_{1i}+ \beta_2x_{2i} + \epsilon_i$. But what is the generally derived formula in this case for $\hat{\beta_1}$ and $\hat{\beta_2}$. How can represent them in a scalar multiplied out value/weight format rather than a matrix/vector format?
 [/INST] Based on the comments above (by @Sycorax):
$$X^TX = 
\begin{pmatrix} 
x_{11} & x_{12} & ... &x_{1n} \\
x_{21} & x_{22} & ... &x_{2n}  
\end{pmatrix} 
\begin{pmatrix} 
x_{11} & x_{21}  \\
x_{12} & x_{22} \\
\vdots &\vdots\\
x_{1n} & x_{2n}
\end{pmatrix} 
= 
\begin{pmatrix}
x_{11}^2 + x_{12}^2 + \:...\: x_{1n}^2 & x_{11}x_{21} + x_{12}x_{22} + \:...\: +x_{1n}x_{2n} \\
x_{11}x_{21} + x_{12}x_{22} + \:...\: +x_{1n}x_{2n} & x_{21}^2 + x_{22}^2 + \:...\: x_{2n}^2 \\ 
\end{pmatrix}
=
\begin{pmatrix}
\sum x_{1i}^2 & \sum x_{1i}x_{2i}\\
\sum x_{1i}x_{2i} & \sum x_{2i}^2 \\ 
\end{pmatrix}$$
$$(X^TX)^{-1} = \dfrac{1}{\sum x_{1i}^2 \sum x_{2i}^2 - (\sum x_{1i}x_{2i})^2}
\begin{pmatrix}
\sum x_{2i}^2 & - \sum x_{1i}x_{2i} \\
- \sum x_{1i}x_{2i} & \sum x_{1i}^2
\end{pmatrix}$$
$$X^T Y = \begin{pmatrix} 
x_{11} & x_{12} & ... &x_{1n} \\
x_{21} & x_{22} & ... &x_{2n}  
\end{pmatrix} 
\begin{pmatrix} 
y_{1}  \\
y_{2} \\
\vdots \\
y_{n} 
\end{pmatrix} 
= 
\begin{pmatrix} 
x_{11}y_1 + x_{12}y_2 + \: ... \: + x_{1n}y_n \\
x_{21}y_1 + x_{22}y_2 + \: ... \: + x_{2n}y_n
\end{pmatrix}
= 
\begin{pmatrix} 
\sum x_{1i}y_i \\
\sum x_{2i}y_i
\end{pmatrix}
$$
$$\therefore \beta = (X^TX)^{-1}X^TY =
\begin{pmatrix} 
\hat{\beta_1} \\
\hat{\beta_2}
\end{pmatrix} 
=
\dfrac{1}{\sum x_{1i}^2 \sum x_{2i}^2 - (\sum x_{1i}x_{2i})^2}\begin{pmatrix} 
\sum x_{2i}^2 \sum x_{1i}y_i - \sum x_{1i}x_{2i} \sum x_{2i}y_i \\
-\sum x_{1i}x_{2i} \sum x_{1i}y_i + \sum x_{1i}^2 \sum x_{2i}y_i 
\end{pmatrix}
$$
</s>