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Let $A=\left[\begin{array}{lll}n & 0 & 0 \\ 0 & n & 0 \\ 0 & 0 & n\end{array}\right]$ and $B=\left[\begin{array}{lll}0 & 0 & n \\ 0 & n & 0 \\ n & 0 & 0\end{array}\right]$. Then, $A^2+B^2+A B=$
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The correct answer is:
$n(2 n l+B)$
$A^2=\left[\begin{array}{ccc}n^2 & 0 & 0 \\ 0 & n^2 & 0 \\ 0 & 0 & n^2\end{array}\right], B^2=\left[\begin{array}{ccc}n^2 & 0 & 0 \\ 0 & n^2 & 0 \\ 0 & 0 & n^2\end{array}\right]$
$A B=\left[\begin{array}{ccc}0 & 0 & n^2 \\ 0 & n^2 & 0 \\ n^2 & 0 & 0\end{array}\right]$
Now, $A^2+B^2+A B$
$\begin{aligned} & =2\left[\begin{array}{ccc}n^2 & 0 & 0 \\ 0 & n^2 & 0 \\ 0 & 0 & n^2\end{array}\right]+\left[\begin{array}{ccc}0 & 0 & n^2 \\ 0 & n^2 & 0 \\ n^2 & 0 & 0\end{array}\right] \\ & =2 n^2\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]+n\left[\begin{array}{lll}0 & 0 & n \\ 0 & n & 0 \\ n & 0 & 0\end{array}\right] \\ & =2 n^2 I+n B=n(2 n I+B)\end{aligned}$
$A B=\left[\begin{array}{ccc}0 & 0 & n^2 \\ 0 & n^2 & 0 \\ n^2 & 0 & 0\end{array}\right]$
Now, $A^2+B^2+A B$
$\begin{aligned} & =2\left[\begin{array}{ccc}n^2 & 0 & 0 \\ 0 & n^2 & 0 \\ 0 & 0 & n^2\end{array}\right]+\left[\begin{array}{ccc}0 & 0 & n^2 \\ 0 & n^2 & 0 \\ n^2 & 0 & 0\end{array}\right] \\ & =2 n^2\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]+n\left[\begin{array}{lll}0 & 0 & n \\ 0 & n & 0 \\ n & 0 & 0\end{array}\right] \\ & =2 n^2 I+n B=n(2 n I+B)\end{aligned}$
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