1. **Problem:** Find the rank of matrix $$A = \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & -4 \\ 0 & 4 & 0 \end{bmatrix}$$. 2. **Recall:** The rank of a matrix is the maximum number of linearly independent rows or columns. 3. **Step 1:** Write matrix $$A$$: $$A = \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & -4 \\ 0 & 4 & 0 \end{bmatrix}$$ 4. **Step 2:** Perform row operations to find the row echelon form. 5. Swap row 1 and row 2 to get a leading nonzero element in the first row: $$\begin{bmatrix} -1 & 0 & -4 \\ 0 & 1 & 0 \\ 0 & 4 & 0 \end{bmatrix}$$ 6. **Step 3:** Use row 2 to eliminate the 4 in row 3, column 2: $$R_3 \to R_3 - 4R_2 = \begin{bmatrix} 0 & 4 & 0 \end{bmatrix} - 4 \times \begin{bmatrix} 0 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$$ 7. The matrix now is: $$\begin{bmatrix} -1 & 0 & -4 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$ 8. **Step 4:** Count nonzero rows: 2 nonzero rows. 9. **Answer:** The rank of matrix $$A$$ is $$2$$.