Question

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

• A. $A^{-1}$ does not exist
• B. $|3\text{Adj}\left(2A\right)|=1728$
• C. Value of $|A|$ is $-1$
• D. There may exist some diagonal elements such that Tr$\left(A\right)$ is zero

Answer 1

Answer: (B), (C)

Value of $|A|$ is $-1$

Since, $A^2=I$
$\Rightarrow A=A^{-1}$
Thus $A$ is an involutory and diagonal matrix,
Let $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right](a,b,c\ne 0)$.
Given, $A^2=I$
$\Rightarrow {\left|A\right|}^2=1\Rightarrow \left|A\right|=-1$
{$\because$ entries are non-positive}
Thus $\left|A\right|=-1$
As $|A|\ne 0,A^{-1}$ exists
$\left|2A\right|=-8\left|3\text{Adj}\right(2A\left)\right|=27\left|\text{Adj}\right(2A\left)\right|=27(-8{)}^2=1728$