Matrix equations

With numbers, $3x = 6$ and $x \cdot 3 = 6$ have the same answer. Take $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. Do $AX = B$ and $XA = B$ have the same solution $X$?