Question

If $A=\left[\begin{array}{c}3\\ 5\\ 2\end{array}\right]$ and B = [1 0 4], verify that (AB)^T = B^T A^T

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}A=\left[\begin{array}{c}3\\ 5\\ 2\end{array}\right] \\ {A}^T=\left[\begin{array}{ccc}3& 5& 2\end{array}\right] \\ B=\left[\begin{array}{ccc}1& 0& 4\end{array}\right] \\ {B}^T=\left[\begin{array}{c}1\\ 0\\ 4\end{array}\right] \\ \mathrm{Now}, \\ AB=\left[\begin{array}{c}3\\ 5\\ 2\end{array}\right]\left[\begin{array}{ccc}1& 0& 4\end{array}\right] \\ \Rightarrow AB=\left[\begin{array}{ccc}3& 0& 12\\ 5& 0& 20\\ 2& 0& 8\end{array}\right] \\ \Rightarrow {\left(AB\right)}^T=\left[\begin{array}{ccc}3& 5& 2\\ 0& 0& 0\\ 12& 20& 8\end{array}\right]...\left(1\right) \\ {B}^T{A}^T=\left[\begin{array}{c}1\\ 0\\ 4\end{array}\right]\left[\begin{array}{ccc}3& 5& 2\end{array}\right] \\ \Rightarrow B^T{A}^T=\left[\begin{array}{ccc}3& 5& 2\\ 0& 0& 0\\ 12& 20& 8\end{array}\right]...\left(2\right) \\ \Rightarrow {\left(AB\right)}^T=B^T{A}^T\left[\mathrm{From}\mathrm{eqs}.\left(1\right)\hspace{0.17em}\mathrm{and}\left(2\right)\right] \end{gathered}$$