Question

use the indicated row operations to change the matrix. replace $r_2$ by $(-2)r_1 + r_2$.$\

$$\begin{bmatrix}1 & 7 & -9 & -5\\\\2 & -9 & 9 & 5\\\\-3 & 3 & -7 & -8\\end{bmatrix}$$

$the row operation $(-2)r_1 + r_2$ results in $\

$$\begin{bmatrix}\\square & \\square & \\square & \\square\\\\\\square & \\square & \\square & \\square\\\\\\square & \\square & \\square & \\square\\end{bmatrix}$$

$

Explanation:

Step1: Identify \( R_1 \) and \( R_2 \)

\( R_1 = [1, 7, -9, -5] \), \( R_2 = [2, -9, 9, 5] \)

Step2: Calculate \((-2)R_1\)

\((-2)R_1 = [-2\times1, -2\times7, -2\times(-9), -2\times(-5)] = [-2, -14, 18, 10]\)

Step3: Add \((-2)R_1\) and \( R_2 \)

New \( R_2 = (-2)R_1 + R_2 = [-2 + 2, -14 + (-9), 18 + 9, 10 + 5] = [0, -23, 27, 15]\)

Step4: Keep \( R_1 \) and \( R_3 \) unchanged

\( R_1 = [1, 7, -9, -5] \), \( R_3 = [-3, 3, -7, -8] \)

Answer:

\[

$$\begin{bmatrix} 1 & 7 & -9 & -5 \\ 0 & -23 & 27 & 15 \\ -3 & 3 & -7 & -8 \end{bmatrix}$$

\]