<s>[INST] Different condition numbers of $\begin{pmatrix} a & b \\ b & c \end{pmatrix}$ 
Let $a,b,c \in \mathbb{R}$ and $A := \begin{pmatrix} a & b \\ b & c \end{pmatrix}$ and $\det(A) \neq 0$.
Find the condition number with respect to the 1-, 2- and $\infty$-norm and discuss for which relationships among $a,b,c$ the corresponding linear systems are well conditioned.

What I've done so far
We have $A^{-1} = \frac{1}{ac - b^2}\begin{pmatrix} c & -b \\ - b & a\end{pmatrix}$ and, therefore,
\begin{align}
\kappa_1(A)
& = \| A \|_1 \| A^{-1} \|_1
= \max\{a + b, b + c\} \cdot\max\left\{\frac{c - b}{ac - b^2}, \frac{a - b}{ac - b^2} \right\} \\
& = \big( b + \max(a, c)\big) \cdot \frac{1}{ac - b^2}\big( \max(a, c) - b\big) \\
& = \frac{b^2 - (\max(a, c))^2}{b^2 - ac}
\end{align}
Note: The above steps only work for $\det(A) > 0$ and $a,c > b$ and are the same for the $\| \cdot \|_1$-norm, since the matrix is symmetric.
Now, I'm not sure to find out for which values of $a,b,c$ this would be small (other than that for $a = c$, it's 1 and for $a = b = c$ and $b = a \ge c$ it's undefined) and especially not how to find this out which out plotting it with a computer.
While plotting I noticed that when |b| gets large, the term is very close to one for a sufficiently large plane around zero (i.e for $|b| = 60$ the term is very similar to 1 for all $(x,y) \in [-25,25]^2$).
For $a = -c$, the term becomes $\frac{b^2 - c^2}{b^2 + c^2} = \frac{2 b^2}{b^2 + c^2} - 1$, because $\max(-c,c) = |c|$.
Also, you can use
\begin{align}\tag{$\star$} \label{eq:star}
\max(x,y) = \frac{x + y + |x - y|}{2} 
\end{align} to obtain
$$
\kappa_1(A)
= \frac{1}{2}\big( 2b + a + c + |a - c|\big)
\left( \frac{a - b}{ac - b^2} + \left| \frac{2c - a - b}{ac - b^2} \right| \right)
$$

Similar problems arise for the next norm:
We have $\kappa_2 = \sqrt{\rho(A^T A) \rho((A A^T)^{-1})}$ and $A^T A = A A^T = A^2$.
Now, we have
\begin{gather}
A^2 = \begin{pmatrix} a^2 + b^2 & b(a + c) \\ b(a + c) & b^2 + c^2 \end{pmatrix} \\
(A A^T)^{-1} = (A^{2})^{-1}
= \frac{1}{(b^2 - ac)^2} \begin{pmatrix} b^2 + c^2 & -b(a + c) \\ -b(a + c) & a^2 + b^2 \end{pmatrix},
\end{gather}
where $\rho(B)$ is the largest eigenvalue of $B$.
Now we obtain the eigenvalues by finding roots of the [characteristic polynomial][1]:
\begin{gather*}
\det(A^2 - t \cdot I_2)
= (a^2 + b^2 - t)(b^2 + c^2 - t) - b^2(a + c)
\overset{!}{=} 0 \\
\implies t_{1,2} = \frac{a^2 + 2b^2 + c^2 \pm(a + c) \sqrt{(a - c)^2 + 4b^2}}{2} 
\end{gather*}
and, [similarly][2]
$$
\rho(A^{-2})
= \max\left\{ \frac{a^2 + 2b^2 + c^2 + \pm (a + c) \sqrt{(a - c)^2 + 4b^2}}{2(b^2 - ac)^2} \right\}
$$
So,
\begin{align}
\kappa_2(A)
& = \sqrt{\max\left\{\frac{a^2 + 2b^2 + c^2 \pm(a + c) \sqrt{(a - c)^2 + 4b^2}}{2}  \right\} \cdot \max\left\{ \frac{a^2 + 2b^2 + c^2 \pm (a + c) \sqrt{(a - c)^2 + 4b^2}}{2(b^2 - ac)^2} \right\} } \\
& = \frac{1}{| b^2 - ac |} \max\left\{ \frac{a^2 + 2b^2 + c^2 \pm (a + c) \sqrt{(a - c)^2 + 4b^2}}{2} \right\} \\
& \overset{\eqref{eq:star}}{=} \frac{a^2 + 2b^2 + c^2 + |a + c|\sqrt{(a - c)^2 + 4b^2}}{2| b^2 - ac |}
\end{align}
How do I simplify this result?
 [/INST] This is not the complete analysis that the instructor requested, but a simplification which can be used to verify certain aspects of the solution.
It is important to note that a matrix is either well-conditioned or it is ill-conditioned. In particular, it does not matter which norm we use to compute the condition number. This follows from the fact that all induced matrix norms are equivalent. 
The 2-norm condition number of the matrix $A$ can be computed from the eigenvalues of the matrix $A^TA$. Specifically, we have $$\kappa_2(A) =\sqrt{\frac{\lambda_{\max}(A^TA)}{\lambda_{\min}(A^TA)}}.$$ We have already seen that $$A^TA = \begin{bmatrix} a^2+b^2 & b(a+c) \\ b(a+c) & a^2 + c^2 \end{bmatrix}.$$ Now by Gershgorin's circle theorem and fact that $A^TA$ is symmetric positive semidefinite, the eigenvalues of $A^TA$ can be found in the interval $$I = [a^2+b^2 - |b(a+c)|,a^2+b^2 + |b(a+c)|] \cap [0, \infty).$$ We observe that $0$ can be an eigenvalue only when $$a^2 + b^2 - |b(a+c)| \leq 0.$$ If $a^2 + b^2 - |b(a+c)| > 0$, then the condition number is bounded by
$$ \kappa_2(A) = \sqrt{\frac{a^2 + b^2 + |b(a+c)|}{a^2 + b^2 - |b(a+c)|}} = \sqrt{\frac{ 1 + \frac{|b(a+c)|}{a^2+b^2}}{1 - \frac{|b(a+c)|}{a^2+b^2}}} $$
We conclude that the matrix $A$ is necessarily well-conditioned when 
$$ |b(a+c)| \ll a^2+b^2$$ and that it can be ill-conditioned when $$|b(a+c)| \geq a^2+b^2$$
A few sanity check are in order. If $c=-a \not = 0$, then our matrix is orthogonal and perfectly conditioned. If $a=b=c$, then our matrix is singular. 
 It is possible to say with certainty that the matrix is ill-conditioned when $$|b| < \min\{|a|,|c|\} \ll \max\{|a|,|c|\}.$$ In this situation, Gershgorin's circles for $A$ are disjoint and it follows that one eigenvalue of $A$ is substantially smaller (in absolute value) than the other. This in turn implies that $A$ is ill-conditioned.
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