Question

If A is 2 × 3 matrix and B is a matrix such that A^T B and BA^T both are defined, then what is the order of B?

Answer 1

Order of A = $2\times 3$
Order of $A^T=3\times 2$

Let order of B = $m\times n$

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}{A}^{T}B\mathrm{and}B{A}^T\mathrm{are}\mathrm{defined} \\ \mathrm{If}{A^T}_{3\times 2}{B}_{m\times n}\mathrm{exists},\mathrm{then}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{columns}\mathrm{in}{A}^T\mathrm{must}\mathrm{be}\mathrm{equal}\mathrm{to}\mathrm{number}\mathrm{of}\mathrm{rows}\mathrm{in}B. \\ \Rightarrow m=2 \\ \mathrm{If}{B}_{m\times n}{A^T}_{3\times 2}\mathit{}\mathrm{exists},\mathrm{then}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{columns}\mathrm{in}B\mathrm{must}\mathrm{be}\mathrm{equal}\mathrm{to}\mathrm{number}\mathrm{of}\mathrm{rows}\mathrm{in}{A}^T. \\ \Rightarrow n=3 \\ \therefore \mathrm{Order}\mathrm{of}B=2\times 3 \end{gathered}$$