Let order of matrix B is m $\times$ n.
AB and AB' both are defined.
$\Rightarrow {\left[\mathit{A}\right]}_{2\times 3}{\left[\mathit{B}\right]}_{\mathit{m}\times \mathit{n}}={\left[\mathit{A}\mathit{B}\right]}_{2\times \mathit{n}}$ and m = 3
Similarly, ${\left[\mathit{A}\right]}_{2\times 3}{\left[\mathit{B}\prime \right]}_{\mathit{n}\times \mathit{m}}={\left[\mathit{A}\mathit{B}\prime \right]}_{2\times \mathit{m}}$ and n = 3
Therefore, the order of matrix B is 3 $\times$ 3.