Let $$A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}$$. The only correct statement about the matrix $$A$$ is

We are given the matrix:

$$ A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix} $$

Let us evaluate the characteristic behavior and properties of this matrix by finding its square, $$ A^2 $$.

Step 1: Matrix Multiplication

To find $$ A^2 $$, we multiply the matrix $$ A $$ by itself:

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

Step 2: Calculating Row by Column Elements

Row 1, Column 1:

$$ (0)(0) + (0)(0) + (-1)(-1) = 1 $$

Row 1, Column 2:

$$ (0)(0) + (0)(-1) + (-1)(0) = 0 $$

Row 1, Column 3:

$$ (0)(-1) + (0)(0) + (-1)(0) = 0 $$

Row 2, Column 1:

$$ (0)(0) + (-1)(0) + (0)(-1) = 0 $$

Row 2, Column 2:

$$ (0)(0) + (-1)(-1) + (0)(0) = 1 $$

Row 2, Column 3:

$$ (0)(-1) + (-1)(0) + (0)(0) = 0 $$

Row 3, Column 1:

$$ (-1)(0) + (0)(0) + (0)(-1) = 0 $$

Row 3, Column 2:

$$ (-1)(0) + (0)(-1) + (0)(0) = 0 $$

Row 3, Column 3:

$$ (-1)(-1) + (0)(0) + (0)(0) = 1 $$

Step 3: Conclusion

Combining all calculated elements yields the Identity matrix:

$$ A^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = I $$

A matrix whose square equals the identity matrix is called an involutory matrix. Since $$ A^2 = I $$, it also implies that the matrix is its own inverse, meaning $$ A = A^{-1} $$.

Therefore, the only correct statement about the matrix $$ A $$ is that $$ A^2 = I $$ (or that $$ A $$ is an involutory matrix).