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We are tasked with finding the row space, dimension of the row space \(\text{dim(rowsp}(A)\)), and the rank of matrix \( A \).

Given matrix \( A = \begin{pmatrix} 
1 & 2 & 1 \\
2 & 0 & 1 \\
3 & 2 & 2
\end{pmatrix} \),

### Steps to Solve:
1. **Row Reduce the Matrix:** We will row reduce matrix \(A\) to row echelon form (REF).
   
   \[
   A = \begin{pmatrix} 
   1 & 2 & 1 \\
   2 & 0 & 1 \\
   3 & 2 & 2
   \end{pmatrix}
   \]

   Perform \(R_2 \leftarrow R_2 - 2R_1\):
   
   \[
   A = \begin{pmatrix}
   1 & 2 & 1 \\
   0 & -4 & -1 \\
   3 & 2 & 2
   \end{pmatrix}
   \]

   Perform \(R_3 \leftarrow R_3 - 3R_1\):

   \[
   A = \begin{pmatrix}
   1 & 2 & 1 \\
   0 & -4 & -1 \\
   0 & -4 & -1
   \end{pmatrix}
   \]

   Perform \(R_3 \leftarrow R_3 - R_2\):

   \[
   A = \begin{pmatrix}
   1 & 2 & 1 \\
   0 & -4 & -1 \\
   0 & 0 & 0
   \end{pmatrix}
   \]

2. **Determine the Row Space:**
   From the row echelon form, the non-zero rows are:
   \[
   \{(1, 2, 1), (0, -4, -1)\}
   \]
   Thus, the row space is spanned by these two vectors.

3. **Dimension of the Row Space (Rank):**
   The number of non-zero rows in the row echelon form is 2. Therefore, the dimension of the row space and the rank of matrix \(A\) is 2.

4. **Conclusion:**
   Based on the row space and rank, the correct answer is:
   \[
   \boxed{\text{rowsp}(A) = \text{span} \{(1, 2, 1), (0, -4, -1)\}, \text{dim(rowsp}(A)) = 2, \text{rank}(A) = 2}
   \]
   
This corresponds to the first answer choice.

Would you like more details or have any questions?

### Related Questions:
1. How do you find the null space of a matrix?
2. What is the difference between row space and column space?
3. How does the rank-nullity theorem relate to the solution of systems of equations?
4. What is the significance of row reduction in solving linear equations?
5. How can you verify if a set of vectors forms a basis for a vector space?

**Tip:** Row reduction is crucial for determining the rank of a matrix and for solving systems of linear equations efficiently.

Find the row space of matrix A = \(\begin{pmatrix} 1 & 2 & 1 \\ 2 & 0 & 1 \\ 3 & 2 & 2 \end{pmatrix}\), dim[\text{rowsp}(A)] and rank(A)

Learn how to find the row space and rank of matrix A = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 0 & 1 \\ 3 & 2 & 2 \end{pmatrix}. This problem covers key linear algebra concepts such as row space, dimension, and rank of a matrix. Suitable for undergraduate students studying Linear Algebra.

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