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Sagot :
To solve this, we will perform the operation [tex]\(-R1 + R2 \rightarrow R2\)[/tex] and determine the resulting matrix.
Let's start with the initial matrix:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 6 \\ 1 & 1 & 1 & -2 \\ 0 & 2 & 1 & 5 \end{array}\right] \][/tex]
We need to apply the operation [tex]\(-R1 + R2 \rightarrow R2\)[/tex]. This means we will subtract the elements of row 1 from the corresponding elements of row 2 to get the new row 2.
Step-by-step, it looks like this:
1. Element in column 1 of R2:
[tex]\[ 1 - 1 = 0 \][/tex]
2. Element in column 2 of R2:
[tex]\[ 1 - 2 = -1 \][/tex]
3. Element in column 3 of R2:
[tex]\[ 1 - 3 = -2 \][/tex]
4. Element in the augmented column of R2:
[tex]\[ -2 - 6 = -8 \][/tex]
The updated matrix is then:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 6 \\ 0 & -1 & -2 & -8 \\ 0 & 2 & 1 & 5 \end{array}\right] \][/tex]
Therefore, the result of the operation [tex]\(-R1 + R2 \rightarrow R2\)[/tex] applied to the given matrix is:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 6 \\ 0 & -1 & -2 & -8 \\ 0 & 2 & 1 & 5 \end{array}\right] \][/tex]
Among the given options, the correct resulting matrix is:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 6 \\ 0 & -1 & -2 & -8 \\ 0 & 2 & 1 & 5 \end{array}\right] \][/tex]
So, the correct option is:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 6 \\ 0 & -1 & -2 & -8 \\ 0 & 2 & 1 & 5 \end{array}\right] \][/tex]
Let's start with the initial matrix:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 6 \\ 1 & 1 & 1 & -2 \\ 0 & 2 & 1 & 5 \end{array}\right] \][/tex]
We need to apply the operation [tex]\(-R1 + R2 \rightarrow R2\)[/tex]. This means we will subtract the elements of row 1 from the corresponding elements of row 2 to get the new row 2.
Step-by-step, it looks like this:
1. Element in column 1 of R2:
[tex]\[ 1 - 1 = 0 \][/tex]
2. Element in column 2 of R2:
[tex]\[ 1 - 2 = -1 \][/tex]
3. Element in column 3 of R2:
[tex]\[ 1 - 3 = -2 \][/tex]
4. Element in the augmented column of R2:
[tex]\[ -2 - 6 = -8 \][/tex]
The updated matrix is then:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 6 \\ 0 & -1 & -2 & -8 \\ 0 & 2 & 1 & 5 \end{array}\right] \][/tex]
Therefore, the result of the operation [tex]\(-R1 + R2 \rightarrow R2\)[/tex] applied to the given matrix is:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 6 \\ 0 & -1 & -2 & -8 \\ 0 & 2 & 1 & 5 \end{array}\right] \][/tex]
Among the given options, the correct resulting matrix is:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 6 \\ 0 & -1 & -2 & -8 \\ 0 & 2 & 1 & 5 \end{array}\right] \][/tex]
So, the correct option is:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 6 \\ 0 & -1 & -2 & -8 \\ 0 & 2 & 1 & 5 \end{array}\right] \][/tex]
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