Question

For the matrices A and B, verify that (AB)′= B'A' where
(i)A=\(\begin{bmatrix}1\\-4\\3\end{bmatrix},\,B=\begin{bmatrix}-1&2&1\end{bmatrix}\)

(ii)A=\(\begin{bmatrix}0\\1\\2\end{bmatrix},\,B=\begin{bmatrix}1&5&7\end{bmatrix}\)

Answer 1

(i)AB=\(\begin{bmatrix}1\\-4\\3\end{bmatrix}\begin{bmatrix}-1&2&1\end{bmatrix}\)=\(\begin{bmatrix}-1&2&1\\4&-8&-4\\-3&6&3\end{bmatrix}\)

therefore (AB)'=\(\begin{bmatrix}-1&4&-3\\2&-8&6\\1&-4&3\end{bmatrix}\)

Now A'=\(\begin{bmatrix}1&-4&3\end{bmatrix}\),B'=\(\begin{bmatrix}-1\\2\\1\end{bmatrix}\)

Therefore B'A'=\(\begin{bmatrix}-1\\2\\1\end{bmatrix}=\begin{bmatrix}-1&4&-3\\2&-8&6\\1&-4&3\end{bmatrix}\)

Hence we verified that:(AB)′= B'A'

---

(ii)AB=\(\begin{bmatrix}0\\1\\2\end{bmatrix}\begin{bmatrix}1&5&7\end{bmatrix}\)=\(\begin{bmatrix}0&0&0\\1&5&7\\2&10&14\end{bmatrix}\)

so (AB)'=\(\begin{bmatrix}0&1&2\\0&5&10\\0&7&14\end{bmatrix}\)

Now A'=\(\begin{bmatrix}0&1&2\end{bmatrix}\),B'=\(\begin{bmatrix}1\\5\\7\end{bmatrix}\)
so B'A'=\(\begin{bmatrix}1\\5\\7\end{bmatrix}\)\(\begin{bmatrix}0&1&2\end{bmatrix}\)=\(\begin{bmatrix}0&1&2\\0&5&10\\0&7&14\end{bmatrix}\)

Hence we verified that(AB)′= B'A'