Which one of the following matrices can be obtained by performing elementary row transformations on the $3 \times 3$ identity matrix?

Any matrix obtained by performing elementary row transformations on the identity matrix is equivalent to the identity matrix, which means it must be non-singular (invertible). Therefore, its determinant must be non-zero.

Evaluating the determinant of the matrix in option (A):
$\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix} = 0$

Evaluating the determinant of the matrix in option (B):
$\begin{vmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 1 & 2 & 1 \end{vmatrix} = 1(3 - 8) - 1(2 - 4) + 1(4 - 3) = -5 + 2 + 1 = -2 \neq 0$

Evaluating the determinant of the matrix in option (C):
$\begin{vmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 2 & 5 & 8 \end{vmatrix} = 1(24 - 20) - 1(16 - 8) + 1(10 - 6) = 4 - 8 + 4 = 0$

Evaluating the determinant of the matrix in option (D):
$\begin{vmatrix} 1 & 1 & 1 \\ -1 & 1 & 2 \\ 0 & 2 & 3 \end{vmatrix} = 1(3 - 4) - 1(-3 - 0) + 1(-2 - 0) = -1 + 3 - 2 = 0$

Since only the matrix in option (B) has a non-zero determinant, it is the only one that can be obtained from the identity matrix by elementary row transformations.

Answer: $\begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 1 & 2 & 1 \end{bmatrix}$