1. **State the problem:** We are given the matrix $$\begin{bmatrix} 1 & -2 & 3 & 1 \\ 2 & -1 & 2 & 2 \\ 3 & 1 & 2 & 3 \end{bmatrix}$$ and we want to reduce it to its Reduced Row Echelon Form (RREF). 2. **Recall the goal:** The RREF of a matrix has leading 1s in each row, zeros below and above these leading 1s, and rows of zeros (if any) at the bottom. 3. **Step 1: Make the first pivot 1 (already 1 in row 1, column 1).** 4. **Step 2: Eliminate entries below the pivot in column 1.** - Replace row 2 with row 2 minus 2 times row 1: $$R_2 = R_2 - 2R_1 = [2, -1, 2, 2] - 2[1, -2, 3, 1] = [2-2, -1+4, 2-6, 2-2] = [0, 3, -4, 0]$$ - Replace row 3 with row 3 minus 3 times row 1: $$R_3 = R_3 - 3R_1 = [3, 1, 2, 3] - 3[1, -2, 3, 1] = [3-3, 1+6, 2-9, 3-3] = [0, 7, -7, 0]$$ Matrix now: $$\begin{bmatrix} 1 & -2 & 3 & 1 \\ 0 & 3 & -4 & 0 \\ 0 & 7 & -7 & 0 \end{bmatrix}$$ 5. **Step 3: Make the pivot in row 2, column 2 equal to 1.** Divide row 2 by 3: $$R_2 = \frac{1}{3} R_2 = [0, 1, -\frac{4}{3}, 0]$$ 6. **Step 4: Eliminate entries above and below the pivot in column 2.** - Replace row 1 with row 1 plus 2 times row 2: $$R_1 = R_1 + 2R_2 = [1, -2, 3, 1] + 2[0, 1, -\frac{4}{3}, 0] = [1, 0, 3 - \frac{8}{3}, 1] = [1, 0, \frac{1}{3}, 1]$$ - Replace row 3 with row 3 minus 7 times row 2: $$R_3 = R_3 - 7R_2 = [0, 7, -7, 0] - 7[0, 1, -\frac{4}{3}, 0] = [0, 0, -7 + \frac{28}{3}, 0] = [0, 0, \frac{7}{3}, 0]$$ Matrix now: $$\begin{bmatrix} 1 & 0 & \frac{1}{3} & 1 \\ 0 & 1 & -\frac{4}{3} & 0 \\ 0 & 0 & \frac{7}{3} & 0 \end{bmatrix}$$ 7. **Step 5: Make the pivot in row 3, column 3 equal to 1.** Multiply row 3 by $\frac{3}{7}$: $$R_3 = \frac{3}{7} R_3 = [0, 0, 1, 0]$$ 8. **Step 6: Eliminate entries above the pivot in column 3.** - Replace row 1 with row 1 minus $\frac{1}{3}$ times row 3: $$R_1 = R_1 - \frac{1}{3} R_3 = [1, 0, \frac{1}{3}, 1] - \frac{1}{3}[0, 0, 1, 0] = [1, 0, 0, 1]$$ - Replace row 2 with row 2 plus $\frac{4}{3}$ times row 3: $$R_2 = R_2 + \frac{4}{3} R_3 = [0, 1, -\frac{4}{3}, 0] + \frac{4}{3}[0, 0, 1, 0] = [0, 1, 0, 0]$$ 9. **Final RREF matrix:** $$\begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}$$ This is the reduced row echelon form of the given matrix.