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If $\mathrm{A}=\left[\begin{array}{ccc}1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}-1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5\end{array}\right]$ then which of the following is/are correct?
$1.$ $A$ and $B$ commute.
$2. A B$ is a null matrix. Select the correct answer using the code given below:
Options:
$1.$ $A$ and $B$ commute.
$2. A B$ is a null matrix. Select the correct answer using the code given below:
Solution:
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Verified Answer
The correct answer is:
2 only
$\mathrm{A}=\left[\begin{array}{ccc}1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3\end{array}\right] ; \mathrm{B}=\left[\begin{array}{ccc}-1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5\end{array}\right]$
$\mathrm{AB}=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$
$\mathrm{BA}=\left[\begin{array}{ccc}-8 & 7 & -10 \\ 48 & -42 & 60 \\ 40 & -35 & 50\end{array}\right]$
as $\mathrm{AB} \neq \mathrm{BA}$
So $\mathrm{A}$ and $\mathrm{B}$ are not commute.
$\mathrm{AB}=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$
$\mathrm{BA}=\left[\begin{array}{ccc}-8 & 7 & -10 \\ 48 & -42 & 60 \\ 40 & -35 & 50\end{array}\right]$
as $\mathrm{AB} \neq \mathrm{BA}$
So $\mathrm{A}$ and $\mathrm{B}$ are not commute.
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