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If A is an orthogonal matrix of order 3 and $\mathrm{B}=\left[\begin{array}{ccc}1 & 2 & 3 \\ -3 & 0 & 2 \\ 2 & 5 & 0\end{array}\right]$, then which of the following is/are correct?
$1.$ $|\mathrm{AB}|=\pm 47$
$2. \mathrm{AB}=\mathrm{BA}$
Select the correct answer using the code given below:
Options:
$1.$ $|\mathrm{AB}|=\pm 47$
$2. \mathrm{AB}=\mathrm{BA}$
Select the correct answer using the code given below:
Solution:
1567 Upvotes
Verified Answer
The correct answer is:
1 only
The determinent of a orthogonal matrix is always $\pm 1$ $|\mathrm{A}|=\pm 1$
$\mathrm{B}=\left[\begin{array}{ccc}1 & 2 & 3 \\ -3 & 0 & 2 \\ 2 & 5 & 0\end{array}\right]$
$|B|=-10-2(-4)+3(-15)$
$=-47$
$|\mathrm{AB}|=|\mathrm{A}||\mathrm{B}|$
$=(\pm 1)(-47)$
$=\pm 47$
$\mathrm{B}=\left[\begin{array}{ccc}1 & 2 & 3 \\ -3 & 0 & 2 \\ 2 & 5 & 0\end{array}\right]$
$|B|=-10-2(-4)+3(-15)$
$=-47$
$|\mathrm{AB}|=|\mathrm{A}||\mathrm{B}|$
$=(\pm 1)(-47)$
$=\pm 47$
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