If A is of order p*q and both AB and BA are defined, then show that B is of the order q*p.

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$$\begin{gathered} \mathrm{Order} \mathrm{of} \mathrm{A} \mathrm{is} \mathrm{p}\times \mathrm{q}. \\ \mathrm{For} \mathrm{AB} \mathrm{to} \mathrm{be} \mathrm{defined} \\ \mathrm{Number} \mathrm{of} \mathrm{rows} \mathrm{in} \mathrm{B}=\mathrm{Number} \mathrm{of} \mathrm{columns} \mathrm{in} \mathrm{A} \\ \mathbf{\Rightarrow }\mathbf{Number}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{rows}\mathbf{ }\mathbf{in}\mathbf{ }\mathbf{B}\mathbf{=}\mathbf{q} \\ \mathrm{For} \mathrm{BA} \mathrm{to} \mathrm{be} \mathrm{defined} \\ \mathrm{Number} \mathrm{of} \mathrm{columns} \mathrm{in} \mathrm{B}=\mathrm{Number} \mathrm{of} \mathrm{rows} \mathrm{in} \mathrm{A} \\ \mathbf{\Rightarrow }\mathbf{Number}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{columns}\mathbf{ }\mathbf{in}\mathbf{ }\mathbf{B}\mathbf{=}\mathbf{p} \\ \mathbf{Hence}\mathbf{ }\mathbf{order}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{B}\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{q}\mathbf{\times }\mathbf{p}\mathbf{ }\left(\mathbf{Hence}\mathbf{ }\mathbf{proved}\right) \end{gathered}$$

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