Compute the determinant of the following elementary matrix. \( \left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right] \) Cimplify your answer.)

To compute the determinant of the given elementary matrix \[ A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \] we can use the formula for the determinant of a \( 3 \times 3 \) matrix, which is given by: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg). \] In this case, the entries correspond to: - \( a = 0, b = 1, c = 0 \) - \( d = 1, e = 0, f = 0 \) - \( g = 0, h = 0, i = 1 \) Plugging these values into the determinant formula, we get: \[ \text{det}(A) = 0 \cdot (0 \cdot 1 - 0 \cdot 0) - 1 \cdot (1 \cdot 1 - 0 \cdot 0) + 0 \cdot (1 \cdot 0 - 0 \cdot 0). \] This simplifies to: \[ \text{det}(A) = 0 - 1 \cdot 1 + 0 = -1. \] Therefore, the determinant of the matrix is \[ \boxed{-1}. \]