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If $A=\left(\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1\end{array}\right)$ and $B=\left(\begin{array}{ccc}-5 & 7 & 1 \\ 1 & -5 & 7 \\ 7 & 1 & -5\end{array}\right)$ then $A B$ is equal to
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The correct answer is:
$18 I_3$
We have $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}-5 & 7 & 1 \\ 1 & -5 & 7 \\ 7 & 1 & -5\end{array}\right]$
$\therefore A B=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1\end{array}\right]\left[\begin{array}{ccc}-5 & 7 & 1 \\ 1 & -5 & 7 \\ 7 & 1 & -5\end{array}\right]$
$A B=\left[\begin{array}{ccc}18 & 0 & 0 \\ 0 & 18 & 0 \\ 0 & 0 & 18\end{array}\right]=18\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
$A B=18 I_3$
$\therefore A B=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1\end{array}\right]\left[\begin{array}{ccc}-5 & 7 & 1 \\ 1 & -5 & 7 \\ 7 & 1 & -5\end{array}\right]$
$A B=\left[\begin{array}{ccc}18 & 0 & 0 \\ 0 & 18 & 0 \\ 0 & 0 & 18\end{array}\right]=18\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
$A B=18 I_3$
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