1. **State the problem:** We are given the matrix \( \begin{bmatrix} -1 & 2 & -1 & 0 \\ 2 & 4 & 4 & 2 \\ 0 & 0 & 1 & 5 \\ 1 & 6 & 3 & 2 \end{bmatrix} \) and need to find its echelon form and rank. 2. **Write the original matrix:** $$ A = \begin{bmatrix} -1 & 2 & -1 & 0 \\ 2 & 4 & 4 & 2 \\ 0 & 0 & 1 & 5 \\ 1 & 6 & 3 & 2 \end{bmatrix} $$ 3. **Goal:** Convert \(A\) to row echelon form using elementary row operations. 4. **Start with row 1 (pivot position in column 1):** Row 1 is \([-1, 2, -1, 0]\). 5. **Eliminate entries below pivot in column 1:** - Replace Row 2 by \(\text{Row 2} + 2 \times \text{Row 1}\): \[ 2 + 2\times(-1) = 2 - 2 = 0, 4 + 2\times 2 = 4 + 4 = 8, 4 + 2\times (-1) = 4 - 2 = 2, 2 + 2\times 0 = 2 \] Row 2 new: \([0,8,2,2]\) - Replace Row 4 by \(\text{Row 4} + 1 \times \text{Row 1}\): \[ 1 + (-1) = 0, 6 + 2 = 8, 3 + (-1) = 2, 2 + 0 = 2 \] Row 4 new: \([0,8,2,2]\) Updated matrix: $$ \begin{bmatrix} -1 & 2 & -1 & 0 \\ 0 & 8 & 2 & 2 \\ 0 & 0 & 1 & 5 \\ 0 & 8 & 2 & 2 \end{bmatrix} $$ 6. **Pivot in row 2, column 2 is 8.** 7. **Eliminate below pivot in column 2 (row 4):** Replace Row 4 by \(\text{Row 4} - \text{Row 2}\): \[ 0 - 0 = 0, 8 - 8 = 0, 2 - 2 = 0, 2 - 2 = 0 \] Row 4 new: \([0,0,0,0]\) Updated matrix: $$ \begin{bmatrix} -1 & 2 & -1 & 0 \\ 0 & 8 & 2 & 2 \\ 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$ 8. **Pivot in row 3, column 3 is 1, no rows below to eliminate.** 9. **Matrix is now in echelon form:** $$ \begin{bmatrix} -1 & 2 & -1 & 0 \\ 0 & 8 & 2 & 2 \\ 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$ 10. **Rank:** The number of nonzero rows is 3, so \(\text{rank}(A) = 3\). **Final answer:** - Echelon form: $$ \begin{bmatrix} -1 & 2 & -1 & 0 \\ 0 & 8 & 2 & 2 \\ 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$ - Rank \(=3\).