Estimate the condition number in the second norm of the matrix An Let $A_{n}$ be a matrix of size $n$ for $n \geq 1$, and a structure:
$$
A_{n}=\left[\begin{array}{cccccccc}
\sqrt{21} & 1 & 0 & 0 & \ldots & 0 & 0 & 0 \\
0 & \sqrt{21} & 1 & 0 & \ldots & 0 & 0 & 0 \\
0 & 0 & \sqrt{21} & 1 & \ldots & 0 & 0 & 0 \\
0 & 0 & 0 & \sqrt{21} & \ldots & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & 0 & \ldots & \sqrt{21} & 1 & 0 \\
0 & 0 & 0 & 0 & \ldots & 0 & \sqrt{21} & 1 \\
0 & 0 & 0 & 0 & \ldots & 0 & 0 & \sqrt{21}
\end{array}\right]
$$
Estimate the condition number in the second norm of the matrix $A_{n}$.
My solutions is as follows
$$
\|A\|_{2}=\sqrt{\lambda_{\max }\left(A^{*} A\right)}=\sigma_{\max }(A) .
$$
where $\sigma_{\max }(A)$ represents the largest singular value of matrix $A$. So
$$
\begin{aligned}
&\left\|A_{n}\right\|_{2}=\sqrt{21} \\
&\left\|A_{n}^{-1}\right\|_{2}=\frac{1}{\sqrt{21}}
\end{aligned}
$$
Condition number
$$
\mu=|| A_{n} \| \cdot|| A_{n}^{-1}||=1
$$
I want to check if this is correct and I'm used to only solving this for symmetric matrices

As you indicated, you essentially need to estimate the eigenvalues of $A^*A$.
Fortunately, in this case it is easy to compute $A^*A$ explicitly.
$$
A^*A=\left[\begin{array}{cccccccc}
22 & \sqrt{21} & 0 & 0 & \ldots & 0 & 0 & 0 \\
\sqrt{21} &22 & \sqrt{21} & 0 & \ldots & 0 & 0 & 0 \\
0 & \sqrt{21} &22 & \sqrt{21} & \ldots & 0 & 0 & 0 \\
0 & 0 & \sqrt{21} &22 & \ldots & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & 0 & \dots &22 & \sqrt{21} & 0 \\
0 & 0 & 0 & 0 & \ldots & \sqrt{21} &22 & \sqrt{21} \\
0 & 0 & 0 & 0 & \ldots & 0 & \sqrt{21} &22
\end{array}\right]
$$
You can now bound the eigenvalues of this matrix using the Gershgorin circle theorem, which tells you that they must be in $[22-2\sqrt{21},22+2\sqrt{21}]$.
Now,
$$
22-2\sqrt{21} = (\sqrt{21} - 1)^2, \text{ and}\\
22+2\sqrt{21} = (\sqrt{21} + 1)^2.
$$
Therefore, an estimate for the condition number of $A$ is
$$
\frac{\sqrt{21} + 1}{\sqrt{21} - 1}.
$$
