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Mathematics Question on Determinants

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

II. A= [0\1\2]\begin{bmatrix}0\\\1\\\2\end{bmatrix},B=[157]\begin{bmatrix}1&5&7\end{bmatrix}

Answer

I. A=[1−4\3]\begin{bmatrix}1\\\\-4\\\3\end{bmatrix},B=[−121]\begin{bmatrix}-1&2&1\end{bmatrix}
AB =\begin{bmatrix}1\\\\-4\\\3\end{bmatrix}$$\begin{bmatrix}-1&2&1\end{bmatrix}
so (AB)'=[−14−3\4−86\1−43]\begin{bmatrix}-1&4&-3\\\4&-8&6\\\1&-4&3\end{bmatrix}
Now A'=[−143]\begin{bmatrix}-1&4&3\end{bmatrix},B'=[−1\2\1]\begin{bmatrix}-1\\\2\\\1\end{bmatrix}
so B'A'=\begin{bmatrix}-1&4&3\end{bmatrix}$$\begin{bmatrix}-1\\\2\\\1\end{bmatrix}
=[−14−3\4−86\1−43]\begin{bmatrix}-1&4&-3\\\4&-8&6\\\1&-4&3\end{bmatrix}

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

II. A=[0\1\2]\begin{bmatrix}0\\\1\\\2\end{bmatrix},B=[157]\begin{bmatrix}1&5&7\end{bmatrix}

AB=\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}

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 have verified that (AB)′=B'A'