Which of the following is (are) NOT the square of a 3 × 3 matrix...

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.