Which of the following is (are) NOT the square of a $3\times 3$ matrix with real entries?

Let \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right]\).Then, \(|A|=-1\)
If \(\mathrm{A}\) is a prefect square of matrix \(A_1\). Then, \(A_1^2=A\)
\(\Rightarrow\left|A_1\right|^2=|A| \Rightarrow\left|A_1\right|^2=-1 \Rightarrow A_1\) cannot be a real matrix
So, option correct.
Let \(R=\left[\begin{array}{ccc}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right]\) be the sqare of a \(3 \times 3\) matrix \(B_1\), Then,
\(B=B_1^2 \Rightarrow\left|B_1\right|^2=|B| \Rightarrow\left|B_1\right|^2=\left|\begin{array}{ccc}
-1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -1
\end{array}\right|=-1\)
So, B cannot be the perfect square of a real matrix.
In option, the given matrix is the identity matrix \(I_3\) such that \(I_3=I_3^2\)
\(\begin{aligned}
&\text { Conisder the matrix }\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -1
\end{array}\right] \text { given in option. Clearly, } \\
&{\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -1
\end{array}\right]=\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & -1 & 0
\end{array}\right]\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & -1 & 0
\end{array}\right]}
\end{aligned}\)
Thus, the matrix given in option is the perfect square of a real matrix.