I have the matrix $A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$
$\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}x = \begin{bmatrix} 1 \\ 6 \end{bmatrix}$, has no solutions
So I want to use least squares regression to find the best approximation.
So I use the normal equation - $A^tAx = A^tb$
$\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}x = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}\begin{bmatrix} 1 \\ 6 \end{bmatrix}$
$\begin{bmatrix} 5 & 10 \\ 10 & 20 \end{bmatrix}x = \begin{bmatrix} 13 \\ 26 \end{bmatrix}$
However this if I reduce the augmented matrix
$\begin{bmatrix} 5 & 10 & | & 13 \\ 10 & 20 & | & 26 \end{bmatrix}$
I end up with
$\begin{bmatrix} 1 & 2 & | & 2.6 \\ 0 & 0 & | & 0 \end{bmatrix}$
Which is obviuosly not the solution. So where am I going wrong?