Rref Example 1D5Da7

1. **Problem Statement:** Find the Reduced Row Echelon Form (RREF) of the matrix $$\begin{bmatrix}1 & 2 & -1 & 8 \\ -3 & -6 & 2 & -11 \\ 2 & 4 & 1 & 7\end{bmatrix}$$. 2. **Formula and Rules:** RREF is a form of a matrix where each leading entry is 1, each leading 1 is the only nonzero entry in its column, and rows with all zeros are at the bottom. 3. **Step 1:** Start with the original matrix: $$\begin{bmatrix}1 & 2 & -1 & 8 \\ -3 & -6 & 2 & -11 \\ 2 & 4 & 1 & 7\end{bmatrix}$$ 4. **Step 2:** Use row operations to create zeros below the leading 1 in the first column. - Add 3 times row 1 to row 2: $$R_2 = R_2 + 3R_1 = [-3 + 3(1), -6 + 3(2), 2 + 3(-1), -11 + 3(8)] = [0, 0, -1, 13]$$ - Subtract 2 times row 1 from row 3: $$R_3 = R_3 - 2R_1 = [2 - 2(1), 4 - 2(2), 1 - 2(-1), 7 - 2(8)] = [0, 0, 3, -9]$$ Matrix now: $$\begin{bmatrix}1 & 2 & -1 & 8 \\ 0 & 0 & -1 & 13 \\ 0 & 0 & 3 & -9\end{bmatrix}$$ 5. **Step 3:** Make the leading coefficient in row 2 a 1 by multiplying row 2 by -1: $$R_2 = -1 \times R_2 = [0, 0, 1, -13]$$ 6. **Step 4:** Use row 2 to eliminate the entry in row 3, column 3: - Subtract 3 times row 2 from row 3: $$R_3 = R_3 - 3R_2 = [0, 0, 3 - 3(1), -9 - 3(-13)] = [0, 0, 0, 30]$$ Matrix now: $$\begin{bmatrix}1 & 2 & -1 & 8 \\ 0 & 0 & 1 & -13 \\ 0 & 0 & 0 & 30\end{bmatrix}$$ 7. **Step 5:** Make the leading coefficient in row 3 a 1 by dividing row 3 by 30: $$R_3 = \frac{1}{30} R_3 = [0, 0, 0, 1]$$ 8. **Step 6:** Use row 3 to eliminate the entry in row 1, column 4 and row 2, column 4: - Row 1: Subtract 8 times row 3: $$R_1 = R_1 - 8R_3 = [1, 2, -1, 8 - 8(1)] = [1, 2, -1, 0]$$ - Row 2: Add 13 times row 3: $$R_2 = R_2 + 13R_3 = [0, 0, 1, -13 + 13(1)] = [0, 0, 1, 0]$$ Matrix now: $$\begin{bmatrix}1 & 2 & -1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$$ 9. **Step 7:** Use row 2 to eliminate the entry in row 1, column 3: - Add row 2 to row 1: $$R_1 = R_1 + R_2 = [1, 2, -1 + 1, 0 + 0] = [1, 2, 0, 0]$$ 10. **Step 8:** The matrix is now in RREF: $$\begin{bmatrix}1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$$ This matrix satisfies all RREF conditions: leading 1s, zeros above and below leading 1s, and rows of zeros at the bottom (none here).