Question

If `A= [[3],[5],[2]]` And B=[1  0   4] , Verify that `(AB)^T=B^TA^T` 

Answer 1

\[Given: \hspace{0.167em} A = \begin{bmatrix}3 \\ 5 \\ 2\end{bmatrix}\]

\[ A^T = \begin{bmatrix}3 & 5 & 2\end{bmatrix}\]

\[\]\[B = \begin{bmatrix}1 & 0 & 4\end{bmatrix}\]

\[\]\[ B^T = \begin{bmatrix}1 \\ 0 \\ 4\end{bmatrix}\]

\[\]\[Now, \]

\[AB = \begin{bmatrix}3 \\ 5 \\ 2\end{bmatrix}\begin{bmatrix}1 & 0 & 4\end{bmatrix}\]

\[ \Rightarrow AB = \begin{bmatrix}3 & 0 & 12 \\ 5 & 0 & 20 \\ 2 & 0 & 8\end{bmatrix}\]

\[ \Rightarrow \left( AB \right)^T = \begin{bmatrix}3 & 5 & 2 \\ 0 & 0 & 0 \\ 12 & 20 & 8\end{bmatrix} . . . \left( 1 \right)\]

\[\]

\[ B^T A^T = \begin{bmatrix}1 \\ 0 \\ 4\end{bmatrix}\begin{bmatrix}3 & 5 & 2\end{bmatrix}\]

\[ \Rightarrow B^T A^T = \begin{bmatrix}3 & 5 & 2 \\ 0 & 0 & 0 \\ 12 & 20 & 8\end{bmatrix} . . . \left( 2 \right)\]

\[\]

\[ \Rightarrow \left( AB \right)^T = B^T A^T \left[ \text{From eqs }. (1) \hspace{0.167em} \text{and (2)} \right]\]