Question

A and B are square matrices of order $3\times 3$, A is an orthogonal matrix and B is a skew symmetric matrix. Which of the following statement is not true.

• A. $\left|A\right|=\pm 1$
• B. $\left|B\right|=0$
• C. $\left|AB\right|=1$
• D. $\left|AB\right|=0$

Answer 1

The correct option is C $\left|AB\right|=1$

If A is orthogonal matrix then $A^T=A$
B is skew symmetric, then $B^T=-B$
If $A=\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|$ or
$A=\left|\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right|$
$B=\left|\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right|$
$\therefore \left|AB\right|\ne 1$