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

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!

The determinant of a $$3\times 3$$ matrix being positive is a necessary and sufficient condition for it to have atleast one real root.
Regarding only the $$yz$$ plane, $$D$$ rotates by 180°. What if you rotate by 90° twice?