Identification of Square of Real Three by Three Matrices

To determine which of the given $3 \times 3$ matrices cannot be written as the square of a $3 \times 3$ matrix with real entries, let $A$ be a $3 \times 3$ matrix with real entries ($A \in \mathbb{R}^{3 \times 3}$).

For any real matrix $A$, its determinant $\det(A)$ is a real number. If a matrix $M$ is equal to $A^2$, then: $\det(M) = \det(A^2) = (\det(A))^2 \ge 0$ because the square of any real number is always non-negative. Therefore, a necessary condition for a real $3 \times 3$ matrix $M$ to be the square of a real matrix is that $\det(M) \ge 0$.

Now, let us analyze each option:

1. Option (A):

$$M = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Here, $\det(M) = 1 \ge 0$. We can explicitly choose the real matrix $A = I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$, for which $A^2 = M$. Thus, $M$ is the square of a real $3 \times 3$ matrix.

1. Option (B):

$$M = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$$

Here, $\det(M) = (1)(1)(-1) = -1 < 0$. Since $\det(M) < 0$, it is impossible for $M$ to equal $A^2$ for any real matrix $A$, as $(\det(A))^2 \ge 0$. Thus, $M$ is NOT the square of a real $3 \times 3$ matrix.

1. Option (C):

$$M = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$$

Here, $\det(M) = (1)(-1)(-1) = 1 \ge 0$. Consider the real matrix:

$$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} \in \mathbb{R}^{3 \times 3}$$

Computing its square:

$$A^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} = M$$

Thus, $M$ is the square of a real $3 \times 3$ matrix.

1. Option (D):

$$M = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$$

Here, $\det(M) = (-1)(-1)(-1) = -1 < 0$. Since $\det(M) < 0$, there exists no real matrix $A$ such that $A^2 = M$. Thus, $M$ is NOT the square of a real $3 \times 3$ matrix.

Hence, the matrices given in options (B) and (D) are NOT the square of a $3 \times 3$ matrix with real entries.