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

Answer 1

The correct option is C $|AB|=1$

If $A$ is orthogonal matrix, then $A^T=A^{-1}$
We know that for an orthogonal matrix, $|A|=\pm 1$
$B$ is a $3\times 3$ skew symmetric, then $B^T=-B$
We know that the determinant of a skew-symmetric matrix is equal to zero if its order is odd.
$\Rightarrow |B|=0$

Now $|AB|=|A||B|=0$