Find the reduced row echelon form of this augmented matrix \( \left[\begin{array}{cccc}1 & -5 & 0 & 150 \\ 0 & 1 & 0 & 400 \\ 6 & -9 & 1 & 250\end{array}\right] \)

To find the **Reduced Row Echelon Form (RREF)** of the given augmented matrix: \[ \left[\begin{array}{cccc} 1 & -5 & 0 & 150 \\ 0 & 1 & 0 & 400 \\ 6 & -9 & 1 & 250 \end{array}\right] \] we will perform a series of row operations to simplify the matrix step by step. ### Step 1: Eliminate the Entry Below the Leading 1 in Column 1 - **Objective:** Make the entry below the leading 1 in the first column (which is 6 in Row 3) equal to 0. - **Operation:** \( \text{Row3} = \text{Row3} - 6 \times \text{Row1} \) \[ \begin{aligned} \text{Row3} &= [6, -9, 1, 250] - 6 \times [1, -5, 0, 150] \\ &= [6 - 6, -9 + 30, 1 - 0, 250 - 900] \\ &= [0, 21, 1, -650] \end{aligned} \] **Updated Matrix:** \[ \left[\begin{array}{cccc} 1 & -5 & 0 & 150 \\ 0 & 1 & 0 & 400 \\ 0 & 21 & 1 & -650 \end{array}\right] \] ### Step 2: Eliminate the Entry Below the Leading 1 in Column 2 - **Objective:** Make the entry below the leading 1 in the second column (which is 21 in Row 3) equal to 0. - **Operation:** \( \text{Row3} = \text{Row3} - 21 \times \text{Row2} \) \[ \begin{aligned} \text{Row3} &= [0, 21, 1, -650] - 21 \times [0, 1, 0, 400] \\ &= [0 - 0, 21 - 21, 1 - 0, -650 - 8400] \\ &= [0, 0, 1, -9050] \end{aligned} \] **Updated Matrix:** \[ \left[\begin{array}{cccc} 1 & -5 & 0 & 150 \\ 0 & 1 & 0 & 400 \\ 0 & 0 & 1 & -9050 \end{array}\right] \] ### Step 3: Eliminate the Entry Above the Leading 1 in Column 2 - **Objective:** Make the entry above the leading 1 in the second column (which is -5 in Row 1) equal to 0. - **Operation:** \( \text{Row1} = \text{Row1} + 5 \times \text{Row2} \) \[ \begin{aligned} \text{Row1} &= [1, -5, 0, 150] + 5 \times [0, 1, 0, 400] \\ &= [1 + 0, -5 + 5, 0 + 0, 150 + 2000] \\ &= [1, 0, 0, 2150] \end{aligned} \] **Final RREF Matrix:** \[ \left[\begin{array}{cccc} 1 & 0 & 0 & 2150 \\ 0 & 1 & 0 & 400 \\ 0 & 0 & 1 & -9050 \end{array}\right] \] ### Conclusion The **Reduced Row Echelon Form** of the given augmented matrix is: \[ \begin{bmatrix} 1 & 0 & 0 & 2150 \\ 0 & 1 & 0 & 400 \\ 0 & 0 & 1 & -9050 \end{bmatrix} \] **Answer:** After row reduction, the RREF is  1 0 0 2150  0 1 0 400  0 0 1 −9050 Written as a matrix, [ [1  0  0  2150],  [0  1  0  400],  [0  0  1 −9050] ]