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Mathematics Question on Matrices

For the matrices A and B, verify that (AB)′= B'A' where
(i)A=[1−4\3], B=[−121]\begin{bmatrix}1\\\\-4\\\3\end{bmatrix},\,B=\begin{bmatrix}-1&2&1\end{bmatrix}

(ii)A=[0\1\2], B=[157]\begin{bmatrix}0\\\1\\\2\end{bmatrix},\,B=\begin{bmatrix}1&5&7\end{bmatrix}

Answer

(i)AB=[1−4\3][−121]\begin{bmatrix}1\\\\-4\\\3\end{bmatrix}\begin{bmatrix}-1&2&1\end{bmatrix}=[−121\4−8−4−363]\begin{bmatrix}-1&2&1\\\4&-8&-4\\\\-3&6&3\end{bmatrix}

therefore (AB)'=[−14−3\2−86\1−43]\begin{bmatrix}-1&4&-3\\\2&-8&6\\\1&-4&3\end{bmatrix}

Now A'=[1−43]\begin{bmatrix}1&-4&3\end{bmatrix},B'=[−1\2\1]\begin{bmatrix}-1\\\2\\\1\end{bmatrix}

Therefore B'A'=[−1\2\1]=[−14−3\2−86\1−43]\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=[0\1\2][157]\begin{bmatrix}0\\\1\\\2\end{bmatrix}\begin{bmatrix}1&5&7\end{bmatrix}=[000\157\21014]\begin{bmatrix}0&0&0\\\1&5&7\\\2&10&14\end{bmatrix}

so (AB)'=[012\0510\0714]\begin{bmatrix}0&1&2\\\0&5&10\\\0&7&14\end{bmatrix}

Now A'=[012]\begin{bmatrix}0&1&2\end{bmatrix},B'=[1\5\7]\begin{bmatrix}1\\\5\\\7\end{bmatrix}
so B'A'=\begin{bmatrix}1\\\5\\\7\end{bmatrix}$$\begin{bmatrix}0&1&2\end{bmatrix}=[012\0510\0714]\begin{bmatrix}0&1&2\\\0&5&10\\\0&7&14\end{bmatrix}

Hence we verified that(AB)′= B'A'