Condition numbers
Original equations:
$$
\left\{ \begin{matrix} x_2 = 1 \\ 2 x_1 - x_2 = 1 \end{matrix} \right.
\qquad \Longleftrightarrow \qquad
\begin{bmatrix} 0 & 1 \\ 2 & -1 \end{bmatrix}
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =
\begin{bmatrix} 1 \\ 1 \end{bmatrix}
$$
Condition number of the original equations:
$$
\begin{vmatrix} 0-\lambda & 1 \\ 2 & -1-\lambda \end{vmatrix} = 0
\qquad \Longleftrightarrow \qquad
\lambda(\lambda+1)-2 = 0
\qquad \Longleftrightarrow \\
(\lambda-1)(\lambda+2) = 0
\qquad \Longleftrightarrow \qquad
\left|\frac{\lambda_{\mbox{max}}}{\lambda_{\mbox{min}}}\right| = 2
$$
Squared equations:
$$
\begin{bmatrix} 0 & 1 \\ 2 & -1 \end{bmatrix}
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =
\begin{bmatrix} 1 \\ 1 \end{bmatrix}
\qquad \Longrightarrow \qquad
\begin{bmatrix} 0 & 2 \\ 1 & -1 \end{bmatrix}
\begin{bmatrix} 0 & 1 \\ 2 & -1 \end{bmatrix}
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =
\begin{bmatrix} 0 & 2 \\ 1 & -1 \end{bmatrix}
\begin{bmatrix} 1 \\ 1 \end{bmatrix}
\\ \Longleftrightarrow \qquad
\begin{bmatrix} 4 & -2 \\ -2 & 2 \end{bmatrix}
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =
\begin{bmatrix} 2 \\ 0 \end{bmatrix}
$$
Condition number of the squared equations:
$$
\begin{vmatrix} 4-\lambda & -2 \\ -2 & 2-\lambda \end{vmatrix} = 0
\qquad \Longleftrightarrow \qquad
(\lambda-4)(\lambda-2)-4 = 0
\qquad \Longleftrightarrow \\
\lambda^2-6\lambda+4=0
\qquad \Longleftrightarrow \qquad
(\lambda-[3+\sqrt{5}])(\lambda-[3-\sqrt{5}]) = 0
\\ \Longleftrightarrow \qquad
\left|\frac{\lambda_{\mbox{max}}}{\lambda_{\mbox{min}}}\right|
= \frac{3+\sqrt{5}}{3-\sqrt{5}} \approx 6.854101961 > 2^2
$$
Normed equations:
$$
\begin{bmatrix} 0 & 1 \\ 2 & -1 \end{bmatrix}
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =
\begin{bmatrix} 1 \\ 1 \end{bmatrix}
\qquad \Longleftrightarrow \qquad
\begin{bmatrix} 0 & 1 \\ 2/\sqrt{5} & -1/\sqrt{5} \end{bmatrix}
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =
\begin{bmatrix} 1 \\ 1/\sqrt{5} \end{bmatrix}
$$
Condition number of the normed equations:
$$
\begin{vmatrix} 0-\lambda & 1 \\ 2/\sqrt{5} & -1/\sqrt{5}-\lambda \end{vmatrix} = 0
\qquad \Longleftrightarrow \qquad
\lambda(\lambda+1/\sqrt{5})-2/\sqrt{5} = 0
\qquad \Longleftrightarrow \\
\left(\lambda+\frac{1}{2\sqrt{5}}\right)^2 = \left(\frac{1}{2\sqrt{5}}\right)^2 + \frac{2}{\sqrt{5}}
\qquad \Longleftrightarrow \qquad
\lambda = - \frac{1}{2\sqrt{5}} \left[ 1 \pm \sqrt{1 + 8\sqrt{5}}\, \right]
\\ \Longleftrightarrow \qquad
\left|\frac{\lambda_{\mbox{max}}}{\lambda_{\mbox{min}}}\right| =
\left|\frac{\sqrt{1 + 8\sqrt{5}}+1}{\sqrt{1 + 8\sqrt{5}}-1}\right| \approx 1.597711618
\\ \Longrightarrow \qquad
\left(\left|\frac{\lambda_{\mbox{max}}}{\lambda_{\mbox{min}}}\right|\right)^2 \approx 2.552682414
$$
Normed and squared equations:
$$
\begin{bmatrix} 0 & 2/\sqrt{5} \\ 1 & -1/\sqrt{5} \end{bmatrix}
\begin{bmatrix} 0 & 1 \\ 2/\sqrt{5} & -1/\sqrt{5} \end{bmatrix}
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =
\begin{bmatrix} 0 & 2/\sqrt{5} \\ 1 & -1/\sqrt{5} \end{bmatrix}
\begin{bmatrix} 1 \\ 1/\sqrt{5} \end{bmatrix}
\qquad \Longleftrightarrow \\
\begin{bmatrix} 4/5 & -2/5 \\ -2/5 & 6/5 \end{bmatrix}
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =
\begin{bmatrix} 2/5 \\ 4/5 \end{bmatrix}
$$
Condition number of the normed and squared equations:
$$
\begin{vmatrix} 4/5-\lambda & -2/5 \\ -2/5 & 6/5-\lambda \end{vmatrix} = 0
\qquad \Longleftrightarrow \qquad
(4/5-\lambda)(6/5-\lambda)-4/25 = 0
\qquad \Longleftrightarrow \\
\lambda^2-2\lambda+4/5 = 0
\qquad \Longleftrightarrow \qquad
(\lambda-[1+1/\sqrt{5}])(\lambda-[1-1/\sqrt{5}]) = 0
\\ \Longleftrightarrow \qquad
\left|\frac{\lambda_{\mbox{max}}}{\lambda_{\mbox{min}}}\right| =
\frac{1+1/\sqrt{5}}{1-1/\sqrt{5}} \approx 2.618033987
$$
Note that there is a (small) discrepancy between $\,2.552682414\,$ and $\,2.618033987\,$.
According to theory, the eigenvalues of $A^T$ are those of $A$ but the eigenvectors are in
general not the same: they are mutually orthogonal (reciprocal base). It is easy to see that
the eigenvalues of $A^2$ are the square of the eigenvalues of $A$, but such is (in general)
not the case for the eigenvalues of $A^TA$.