Question

If A is a $3^{rd}$ order square diagonal matrix of non-positive entries such that $A^2=I$, then

• A. There may exist some diagonal element in $A$ which is $zero$
• B. Value of $|A|$ is $-1$
• C. $A^{-1}$ does not exist
• D. $|3adj\left(2A\right)|=1728$

Answer 1

Answer: (B), (D)

The correct options are
B Value of $|A|$ is $-1$
D $|3adj\left(2A\right)|=1728$

If A is an involutor and diagonal matrix, then $A^2=I$.
Let $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$. Then $A^2=I$
$\Rightarrow |A{|}^2=1$ $\Rightarrow |A|=-1$ {$\because$ entries are non-positive}
Thus $|A|=-1$
$A^{-1}=A$
$|2A|=-8$
$|3adj\left(2A\right)|=27.|adj\left(2A\right)|=27(-8{)}^2=1728$