Which of the following is (are) NOT the square of a `3xx3` matrix with real entries? `[1 0 0 0 1 0 0 0-1]` (b) `[-1 0 0 0-1 0 0 0-1]` `[1 0 0 0 1 0 0 0 1]` (d) `[1 0 0 0-1 0 0 0-1]`

Options

A. `{:[(1,0,0),(0,1,0),(0,0,-1)]:}`
B. `{:[(-1,0,0),(0,-1,0),(0,0,-1)]:}`
C. `{:[(1,0,0),(0,1,0),(0,0,1)]:}`
D. `{:[(-1,0,0),(0,-1,0),(0,0,-1)]:}`

Explanation

To determine which of the given matrices is NOT the square of a \(3 \times 3\) matrix with real entries, we will analyze the determinants of each matrix. The key point to remember is that the determinant of a square matrix must be non-negative if it is the square of another matrix.

### Step-by-Step Solution:

1. **Matrix A1:**
\[ A_1 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \]
**Calculate the determinant:**
\[ \text{det}(A_1) = 1 \cdot (1 \cdot (-1) - 0) - 0 + 0 = -1 \]
Since the determinant is negative, \(A_1\) cannot be the square of a \(3 \times 3\) matrix.

2. **Matrix A2:**
\[ A_2 = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \]
**Calculate the determinant:**
\[ \text{det}(A_2) = (-1) \cdot (-1) \cdot (-1) = -1 \]
Again, since the determinant is negative, \(A_2\) cannot be the square of a \(3 \times 3\) matrix.

3. **Matrix A3:**
\[ A_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
**Calculate the determinant:**
\[ \text{det}(A_3) = 1 \cdot (1 \cdot 1 - 0) - 0 + 0 = 1 \]
The determinant is positive, so \(A_3\) can be the square of a \(3 \times 3\) matrix.

4. **Matrix A4:**
\[ A_4 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \]
**Calculate the determinant:**
\[ \text{det}(A_4) = 1 \cdot (-1) \cdot (-1) = 1 \]
The determinant is positive, so \(A_4\) can also be the square of a \(3 \times 3\) matrix.

### Conclusion:
The matrices that are NOT the square of a \(3 \times 3\) matrix with real entries are:
- \(A_1\) and \(A_2\).