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).

   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.