Row reduction of matrix A:
Given matrix:

Apply elementary row operations:

• $R_2 \to R_2 - 2R_1$:
  $$R_2 = [2-2, 6-4, -3-0, -3 - (-2)] = [0, 2, -3, -1]$$
• $R_3 \to R_3 - 3R_1$:
  $$R_3 = [3-3, 10-6, -6-0, -5 - (-3)] = [0, 4, -6, -2]$$
• Now observe $R_3 = 2 R_2$. Apply $R_3 \to R_3 - 2R_2$:
  $$R_3 = [0-0, 4-4, -6 - (-6), -2 - (-2)] = [0, 0, 0, 0]$$
  The row echelon form is:
  $$\begin{bmatrix} 1 & 2 & 0 & -1 \\ 0 & 2 & -3 & -1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$
  There are exactly 2 non-zero linearly independent rows. Thus, $\text{rank}(A) = 2$.
• Incorrect: Rank 0 applies only to the null matrix.
• Correct: Rank is exactly 2.
• Incorrect: Rank cannot exceed the row count (3).
• Incorrect: Row 3 is a linear combination of Row 1 and Row 2, so rank cannot be 3.