Get expert advice and insights on any topic with IDNLearn.com. Find the solutions you need quickly and accurately with help from our knowledgeable community.
Sagot :
Sure, let's go through the detailed steps to perform the row operation [tex]\( R_2 = -9R_1 + R_2 \)[/tex] on the given augmented matrix:
[tex]\[ \left[\begin{array}{rrr|r} 1 & 1 & -9 & -4 \\ 9 & 6 & 8 & 1 \\ -5 & 6 & 7 & 4 \end{array}\right] \][/tex]
### Step 1: Identify the Row Operation
We're instructed to perform the row operation on the second row [tex]\( R_2 \)[/tex]:
[tex]\[ R_2 = -9R_1 + R_2 \][/tex]
### Step 2: Multiply [tex]\( R_1 \)[/tex] by [tex]\(-9\)[/tex]
First, we need to multiply each element of the first row ([tex]\( R_1 \)[/tex]) by [tex]\(-9\)[/tex]:
[tex]\[ R_1 = \left[\begin{array}{rrrr} 1 & 1 & -9 & -4 \end{array}\right] \][/tex]
[tex]\[ -9 \times R_1 = \left[\begin{array}{rrrr} -9 \times 1 & -9 \times 1 & -9 \times (-9) & -9 \times (-4) \end{array}\right] = \left[\begin{array}{rrrr} -9 & -9 & 81 & 36 \end{array}\right] \][/tex]
### Step 3: Add the Result to [tex]\( R_2 \)[/tex]
Now, we add this result to the elements of the second row [tex]\( R_2 \)[/tex]:
[tex]\[ R_2 = \left[\begin{array}{rrrr} 9 & 6 & 8 & 1 \end{array}\right] \][/tex]
[tex]\[ R_2 = -9R_1 + R_2 = \left[\begin{array}{rrrr} -9 & -9 & 81 & 36 \end{array}\right] + \left[\begin{array}{rrrr} 9 & 6 & 8 & 1 \end{array}\right] \][/tex]
Combining the elements, we get:
[tex]\[ R_2 = \left[\begin{array}{rrrr} -9 + 9 & -9 + 6 & 81 + 8 & 36 + 1 \end{array}\right] = \left[\begin{array}{rrrr} 0 & -3 & 89 & 37 \end{array}\right] \][/tex]
### Step 4: Substitute Back into the Matrix
Finally, we substitute the new [tex]\( R_2 \)[/tex] back into the original matrix:
[tex]\[ \left[\begin{array}{rrr|r} 1 & 1 & -9 & -4 \\ 0 & -3 & 89 & 37 \\ -5 & 6 & 7 & 4 \end{array}\right] \][/tex]
Thus, after performing the row operation [tex]\( R_2 = -9R_1 + R_2 \)[/tex], the augmented matrix becomes:
[tex]\[ \left[\begin{array}{rrr|r} 1 & 1 & -9 & -4 \\ 0 & -3 & 89 & 37 \\ -5 & 6 & 7 & 4 \end{array}\right] \][/tex]
[tex]\[ \left[\begin{array}{rrr|r} 1 & 1 & -9 & -4 \\ 9 & 6 & 8 & 1 \\ -5 & 6 & 7 & 4 \end{array}\right] \][/tex]
### Step 1: Identify the Row Operation
We're instructed to perform the row operation on the second row [tex]\( R_2 \)[/tex]:
[tex]\[ R_2 = -9R_1 + R_2 \][/tex]
### Step 2: Multiply [tex]\( R_1 \)[/tex] by [tex]\(-9\)[/tex]
First, we need to multiply each element of the first row ([tex]\( R_1 \)[/tex]) by [tex]\(-9\)[/tex]:
[tex]\[ R_1 = \left[\begin{array}{rrrr} 1 & 1 & -9 & -4 \end{array}\right] \][/tex]
[tex]\[ -9 \times R_1 = \left[\begin{array}{rrrr} -9 \times 1 & -9 \times 1 & -9 \times (-9) & -9 \times (-4) \end{array}\right] = \left[\begin{array}{rrrr} -9 & -9 & 81 & 36 \end{array}\right] \][/tex]
### Step 3: Add the Result to [tex]\( R_2 \)[/tex]
Now, we add this result to the elements of the second row [tex]\( R_2 \)[/tex]:
[tex]\[ R_2 = \left[\begin{array}{rrrr} 9 & 6 & 8 & 1 \end{array}\right] \][/tex]
[tex]\[ R_2 = -9R_1 + R_2 = \left[\begin{array}{rrrr} -9 & -9 & 81 & 36 \end{array}\right] + \left[\begin{array}{rrrr} 9 & 6 & 8 & 1 \end{array}\right] \][/tex]
Combining the elements, we get:
[tex]\[ R_2 = \left[\begin{array}{rrrr} -9 + 9 & -9 + 6 & 81 + 8 & 36 + 1 \end{array}\right] = \left[\begin{array}{rrrr} 0 & -3 & 89 & 37 \end{array}\right] \][/tex]
### Step 4: Substitute Back into the Matrix
Finally, we substitute the new [tex]\( R_2 \)[/tex] back into the original matrix:
[tex]\[ \left[\begin{array}{rrr|r} 1 & 1 & -9 & -4 \\ 0 & -3 & 89 & 37 \\ -5 & 6 & 7 & 4 \end{array}\right] \][/tex]
Thus, after performing the row operation [tex]\( R_2 = -9R_1 + R_2 \)[/tex], the augmented matrix becomes:
[tex]\[ \left[\begin{array}{rrr|r} 1 & 1 & -9 & -4 \\ 0 & -3 & 89 & 37 \\ -5 & 6 & 7 & 4 \end{array}\right] \][/tex]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Find precise solutions at IDNLearn.com. Thank you for trusting us with your queries, and we hope to see you again.