$C^*$-algebra of matrices over $C(\mathcal{X})$.

Theorem \[thm\_invert\] then states that $A$ is invertible if and only if the corresponding curve $\mathcal{K} = \{A(t)\,;\, t\in [0,1]\}$ does not intersect the set of singular $n\times n$ complex matrices, i.e., if and only if $\mathcal{K}$ is entirely contained in the group $\mathcal{GL}_n(\mathbb{C})$ of invertible $n\times n$ complex matrices.

As a simple illustration of Theorem \[thm\_matricial\_norm\], consider the $2\times 2$ Hermitian matrix over $C([0,1])$: $$A = A(t) = \left [ \begin{array}{cc} 
               {\rm e}^{-t} & t^2 + 1 \\
               t^2 + 1 & {\rm e}^t
\end{array}
\right ]\,.$$ Observing now that $$\hat A =  \left [ \begin{array}{cc}
               1 & 2  \\
               2 & {\rm e}
\end{array}
\right ]\,,$$ we obtain the bound $\rho(A) = \|A\|\le \|\hat A\|_2 \approx 4.03586$. By direct computation we find that ${\rm det}(\lambda I_2 - A(t)) = \lambda^2 -2 (\cosh t) \lambda - t^2(t^2+ 2)$ and thus the spectrum of $A(t)$ (as an element of the $C^*$-algebra of matrices over $C([0,1])$) consists of all the numbers of the form $$\lambda_{\pm} (t)= \cosh t \pm \sqrt{\cosh^2 t + t^2(t^2 + 2)}, 
\quad 0\le t \le 1$$ (a compact subset of $\mathbb{R}$). Also note that ${\rm det}(A(t)) = - t^2(t^2 + 2)$ vanishes 