Is $S=\{f:[0,1]\to M_2(\mathbb{C}): f$ continuous and $ f(0)=(\begin{smallmatrix} 0 & 0 \\ 0 & 0 \end{smallmatrix})\}$ commutative?

Question

Let $S=\{f:[0,1]\to M_2(\mathbb{C}): f$ continuous and $ f(0)=\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\}$. Here is $M_2(\mathbb{C})$ endowed with the operator norm from $L(\mathbb{C}^2)$ and $S$ endowed with the pointwise multiplication. My question: Are in $S$ only diagonal matrices? Or why is $S$ commutative?

Answer 1

Consider $f(t)=\begin{pmatrix} t & 0 \\ 0 & 0 \end{pmatrix}$ and $g(t) = \begin{pmatrix} 0 & 0 \\ t & 0 \end{pmatrix}$. Then $f,g$ lie in $S$ and they do not commute.