In $M_n(\mathbb{R})$, does the tangent vector to a path commute with the position?

Question

Let $A_t\in M_n(\mathbb{R})$ be a smooth path. Does the commutator $[A_t',A_t]$ (always) vanish ?

Answer 1

Suppose $[A,B]\neq 0$, write $A_t=A+tB$, $A'_t=B$.

$A_tA'_t=(A+tB)B=AB+tB^2$

$A'_tA_t=B(A+tB)=BA+tB^2$

implies $[A_t,A'_t]\neq 0$.