Why can't matrix commute? Sorry if this question sounds silly, and it probably is. I can prove easily that matrix multiplication is noncommutative; however, look at this 'fake' proof:
$$AB=e^{\ln(A)}e^{\ln(B)}=e^{\ln(A)+\ln(B)}$$
$\ln(A)$ and $\ln(B)$ are also matrices, and matrix addition is comutative. So:
$$e^{\ln(A)+\ln(B)}=e^{\ln(B)+\ln(A)}=e^{\ln(B)}e^{\ln(A)}=BA$$
What's the problem?

Here is a simple example you can try by hand.
Let
$$
A=\begin{bmatrix} 1&0\\0&0\end{bmatrix},\qquad\qquad B=\begin{bmatrix} 0&1\\0&0\end{bmatrix} . 
$$
Then
$$
e^A=\begin{bmatrix} e&0\\0&1\end{bmatrix},\qquad \qquad e^B=\begin{bmatrix} 1&1\\0&1\end{bmatrix},
$$
$\ $
$$
e^Ae^B=\begin{bmatrix} e&e\\0&1\end{bmatrix},\qquad\qquad e^Be^A=\begin{bmatrix} e&1\\0&1\end{bmatrix} \qquad\qquad e^{A+B}=\begin{bmatrix} e&e-1\\0&1\end{bmatrix}.
$$
