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. A=\(\begin{bmatrix}1\\-4\\3\end{bmatrix}\),B=\(\begin{bmatrix}-1&2&1\end{bmatrix}\)
AB =\(\begin{bmatrix}1\\-4\\3\end{bmatrix}\)\(\begin{bmatrix}-1&2&1\end{bmatrix}\)
so (AB)'=\(\begin{bmatrix}-1&4&-3\\4&-8&6\\1&-4&3\end{bmatrix}\)
Now A'=\(\begin{bmatrix}-1&4&3\end{bmatrix}\),B'=\(\begin{bmatrix}-1\\2\\1\end{bmatrix}\)
so B'A'=\(\begin{bmatrix}-1&4&3\end{bmatrix}\)\(\begin{bmatrix}-1\\2\\1\end{bmatrix}\)
=\(\begin{bmatrix}-1&4&-3\\4&-8&6\\1&-4&3\end{bmatrix}\)

Hence we have verified that (AB)′=B'A' 

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

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'=\(\begin{bmatrix}1\\5\\7\end{bmatrix}\)

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 have verified that (AB)′=B'A'