Reduce to reduced row echelon form. \( \left[\begin{array}{ccc}4 & 1 & 21 \\ 2 & 5 & 15 \\ -2 & -1 & -11\end{array}\right] \)

To reduce the matrix to reduced row echelon form (RREF), we can use row operations. 1. Start with the original matrix: \[ \left[\begin{array}{ccc}4 & 1 & 21 \\ 2 & 5 & 15 \\ -2 & -1 & -11\end{array}\right] \] 2. Make the leading coefficient of the first row a 1 by dividing the entire row by 4: \[ R_1 = \frac{1}{4}R_1 \implies R_1 = \left[\begin{array}{ccc}1 & \frac{1}{4} & \frac{21}{4}\end{array}\right] \] Matrix becomes: \[ \left[\begin{array}{ccc}1 & \frac{1}{4} & \frac{21}{4} \\ 2 & 5 & 15 \\ -2 & -1 & -11\end{array}\right] \] 3. Eliminate the first column values below the leading 1 in row 1. Use row 1 to update rows 2 and 3: - For row 2: \(R_2 = R_2 - 2R_1 \) \[ R_2 = \left[\begin{array}{ccc}0 & \frac{19}{4} & \frac{-3}{4}\end{array}\right] \] - For row 3: \(R_3 = R_3 + 2R_1\) \[ R_3 = \left[\begin{array}{ccc}0 & \frac{1}{2} & \frac{1}{2}\end{array}\right] \] Matrix becomes: \[ \left[\begin{array}{ccc}1 & \frac{1}{4} & \frac{21}{4} \\ 0 & \frac{19}{4} & -\frac{3}{4} \\ 0 & \frac{1}{2} & \frac{1}{2}\end{array}\right] \] 4. Next, make the leading coefficient in row 2 a 1 by multiplying by \(\frac{4}{19}\): \[ R_2 = \frac{4}{19}R_2 \implies R_2 = \left[\begin{array}{ccc}0 & 1 & -\frac{3}{19}\end{array}\right] \] Matrix becomes: \[ \left[\begin{array}{ccc}1 & \frac{1}{4} & \frac{21}{4} \\ 0 & 1 & -\frac{3}{19} \\ 0 & \frac{1}{2} & \frac{1}{2}\end{array}\right] \] 5. Eliminate the values above and below 1 in column 2. Use row 2 to update rows 1 and 3: - For row 1: \(R_1 = R_1 - \frac{1}{4}R_2\) \[ R_1 = \left[\begin{array}{ccc}1 & 0 & \frac{27}{19}\end{array}\right] \] - For row 3: \(R_3 = R_3 - \frac{1}{2}R_2\) \[ R_3 = \left[\begin{array}{ccc}0 & 0 & \frac{11}{38}\end{array}\right] \] Matrix becomes: \[ \left[\begin{array}{ccc}1 & 0 & \frac{27}{19} \\ 0 & 1 & -\frac{3}{19} \\ 0 & 0 & \frac{11}{38}\end{array}\right] \] 6. Now, normalize row 3 by multiplying by \(\frac{38}{11}\): \[ R_3 = \frac{38}{11}R_3 \implies R_3 = \left[\begin{array}{ccc}0 & 0 & 1\end{array}\right] \] Matrix becomes: \[ \left[\begin{array}{ccc}1 & 0 & \frac{27}{19} \\ 0 & 1 & -\frac{3}{19} \\ 0 & 0 & 1\end{array}\right] \] 7. Lastly, eliminate the values in column 3 from