Question

If $A=\left[\begin{array}{ccc}a& b& c\\ x& y& z\\ p& q& r\end{array}\right]$, $B=\left[\begin{array}{ccc}q& -b& y\\ -p& a& -x\\ r& -c& z\end{array}\right]$ then

• A. $|A|=|B|$
• B. $|A|=-|B|$
• C. $|A|=2|B|$
• D. $A$ is invertible if and only if $B$ is invertible.

Answer 1

Answer: (B), (D)

The correct options are
B $A$ is invertible if and only if $B$ is invertible.
D $|A|=-|B|$

$|B|=\left|\begin{array}{ccc}q& -b& y\\ -p& a& -x\\ r& -c& z\end{array}\right|=(-1)\left|\begin{array}{ccc}q& -b& y\\ p& -a& x\\ r& -c& z\end{array}\right|$

$=\left|\begin{array}{ccc}q& b& y\\ p& a& x\\ r& c& z\end{array}\right|=-\left|\begin{array}{ccc}p& a& x\\ q& b& y\\ r& c& z\end{array}\right|$

$=\left|\begin{array}{ccc}a& p& x\\ b& q& y\\ c& r& z\end{array}\right|=-\left|\begin{array}{ccc}a& x& p\\ b& y& q\\ c& z& r\end{array}\right|$

$=-|A|$ [$\because |A^T|=|A|$]

$\therefore$ From here, we get $|A|\ne 0$ if and only if $|B|\ne 0$.
i.e $A$ is invertible if and only if $B$ is invertible.

Hence, options B and D.