Checking if given matrix is perfect square of another matrix with real entries

Question

The question is from JEE Advanced(2017), where it asks to identify matrices which are the square of a matrix with real entries: I first found out the determinant of all the matrices. Options (A & B) both have negative determinant value and hence can't be expressed as square of a matrix with real entries. For explanation let any of the two be called as $A$ . Given they are square of another matrix(let $B$) so $B^2=A$ . Taking the determinant on both sides, I get $|B|^2=|A|$ and since $|A|$ is negative, I get $|B|^2<0$. So $B$ can't have all real entries. Option C is $I$ whose square is $I$ or vice versa. I am having trouble with option D . Since it's determinant is also positive and I can't find a simple matrix which when squared gives that option. I have talked to my teacher . He said there is a method to find square root of a matrix but that's far beyond our level. I am a High school student studying in grade 12. So please, if possible, give a simplified hint/answer. Thanks in advance!

Answer 1

The determinant of a $3\times 3$ matrix being positive is a necessary and sufficient condition for it to have atleast one real root.

Answer 2

Regarding only the $yz$ plane, $D$ rotates by 180°. What if you rotate by 90° twice?