Question

If A is an orthogonal matrix of order 3 and $B=\left[\begin{array}{ccc}1& 2& 3\\ -3& 0& 2\\ 2& 5& 0\end{array}\right]$, then which of the following is/are correct?
1. $|AB|=\pm 47$
2. $AB=BA$
Select the correct answer using the code given below :

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

Answer: (A)

1 only

Solution:

$A$ is an orthogonal matrix.

$A^2=1$

$|A{|}^2=1$

$|A|=\pm 1$

$|AB|=\pm |B|$

Now, $|B|=\left[\begin{array}{ccc}1& 2& 3\\ -3& 0& 2\\ 2& 5& 0\end{array}\right]=47$

$\therefore |AB|=\pm 47$

And $AB=\left[\begin{array}{ccc}3& 7& 3\\ -3& 0& 2\\ 2& 5& 0\end{array}\right]$ and

$BA=\left[\begin{array}{ccc}1& 2& 3\\ -3& 0& 2\\ 2& 5& 0\end{array}\right]$

Since, $AB\ne BA$

Hence, A is the correct option.