1. **Problem Statement:** Find the rank of matrix $A = \begin{bmatrix}1 & 2 & 3 \\ 3 & 4 & 5 \\ 3 & 2 & 1\end{bmatrix}$. 2. **Recall:** The rank of a matrix is the maximum number of linearly independent rows or columns. 3. **Step 1:** Write matrix $A$: $$A = \begin{bmatrix}1 & 2 & 3 \\ 3 & 4 & 5 \\ 3 & 2 & 1\end{bmatrix}$$ 4. **Step 2:** Use row operations to find the echelon form. - Replace row 2 by row 2 minus 3 times row 1: $$R_2 = R_2 - 3R_1 = \begin{bmatrix}3 - 3(1) & 4 - 3(2) & 5 - 3(3)\end{bmatrix} = \begin{bmatrix}0 & -2 & -4\end{bmatrix}$$ - Replace row 3 by row 3 minus 3 times row 1: $$R_3 = R_3 - 3R_1 = \begin{bmatrix}3 - 3(1) & 2 - 3(2) & 1 - 3(3)\end{bmatrix} = \begin{bmatrix}0 & -4 & -8\end{bmatrix}$$ 5. **Step 3:** Now matrix looks like: $$\begin{bmatrix}1 & 2 & 3 \\ 0 & -2 & -4 \\ 0 & -4 & -8\end{bmatrix}$$ 6. **Step 4:** Replace row 3 by row 3 minus 2 times row 2: $$R_3 = R_3 - 2R_2 = \begin{bmatrix}0 & -4 - 2(-2) & -8 - 2(-4)\end{bmatrix} = \begin{bmatrix}0 & 0 & 0\end{bmatrix}$$ 7. **Step 5:** The matrix in echelon form is: $$\begin{bmatrix}1 & 2 & 3 \\ 0 & -2 & -4 \\ 0 & 0 & 0\end{bmatrix}$$ 8. **Step 6:** Count nonzero rows: 2. **Final answer:** The rank of matrix $A$ is $2$.