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

The correct option is:
$\left[\begin{array}{ccc}1& 1& 1\\ 2& 3& 4\\ 1& 2& 1\end{array}\right]$

Understanding Elementary Row Transformations:
Elementary row operations do not change the rank or the determinant (up to a non-zero scalar factor) of a matrix. Specifically, a matrix can be obtained by performing elementary row operations on the 3 × 3 identity matrix I if and only if it is row equivalent to I. A matrix is row equivalent to the identity matrix if and only if it is non-singular, i.e., its determinant is non-zero (det(A) ≠ 0).

Let us evaluate the determinant of each given option matrix to check which matrix is non-singular.

Step 1: Check the first option matrix
$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$

Since all rows are identical, det(A) = 0. Therefore, it cannot be obtained from the identity matrix.

Step 2: Check the second option matrix
$B=\left[\begin{array}{ccc}1& 1& 1\\ 2& 3& 4\\ 1& 2& 1\end{array}\right]$

Let us calculate the determinant of matrix B:
det(B) = 1 · (3 · 1 - 4 · 2) - 1 · (2 · 1 - 4 · 1) + 1 · (2 · 2 - 3 · 1)
det(B) = 1 · (3 - 8) - 1 · (2 - 4) + 1 · (4 - 3)
det(B) = 1 · (-5) - 1 · (-2) + 1 · (1)
det(B) = -5 + 2 + 1 = -2

Since det(B) = -2 ≠ 0, matrix B is non-singular and has rank 3. Hence, it can be reduced to the identity matrix using elementary row operations, and conversely, it can be obtained from the 3 × 3 identity matrix via elementary row operations.

Step 3: Verify the remaining options for completeness
For the third option matrix:
$C=\left[\begin{array}{ccc}1& 1& 1\\ 2& 3& 4\\ 2& 5& 8\end{array}\right]$

Notice that Row 3 is a linear combination of Row 1 and Row 2: Row 3 = 2 · Row 2 - Row 1 = [4-1, 6-1, 8-1] = [3, 5, 7], or let us directly calculate its determinant:
det(C) = 1 · (24 - 20) - 1 · (16 - 8) + 1 · (10 - 6) = 4 - 8 + 4 = 0.
Thus, det(C) = 0.

For the fourth option matrix:
$D=\left[\begin{array}{ccc}1& 1& 1\\ -1& 1& 2\\ 0& 2& 3\end{array}\right]$

Adding Row 1 to Row 2 gives [0, 2, 3], which makes Row 2 identical to Row 3. Therefore, det(D) = 0.

Conclusion:
Only the second matrix has a non-zero determinant, meaning it is non-singular and can be obtained by performing elementary row transformations on the 3 × 3 identity matrix.