To determine the number by which row 2 of the matrix
[tex]\[
\left[\begin{array}{ccc:c}
1 & 1 & 1 & 6 \\
2 & 0 & -2 & -14 \\
1 & 1 & 0 & 3
\end{array}\right]
\][/tex]
should be multiplied to obtain the new matrix
[tex]\[
\left[\begin{array}{ccc:c}
1 & 1 & 1 & 6 \\
1 & 0 & -1 & -7 \\
1 & 1 & 0 & 3
\end{array}\right],
\][/tex]
we need to compare the elements of the second row in both matrices.
The initial second row is [tex]\([2, 0, -2, -14]\)[/tex].
The target second row is [tex]\([1, 0, -1, -7]\)[/tex].
To find the multiplier, we can check how each element in the initial row transforms to the corresponding element in the target row.
Let's look at the elements one by one:
1. The first element in the initial row is 2, and it should become 1.
[tex]\[
\frac{1}{2} = 0.5
\][/tex]
2. The second element in the initial row is 0, and it should remain 0. This tells us that the multiplication factor works as expected since [tex]\(0 \times 0.5 = 0\)[/tex].
3. The third element in the initial row is -2, and it should become -1.
[tex]\[
\frac{-1}{-2} = 0.5
\][/tex]
4. The fourth element in the initial row is -14, and it should become -7.
[tex]\[
\frac{-7}{-14} = 0.5
\][/tex]
Since all elements share a common factor of 0.5, we conclude that the multiplier that transforms the second row of the initial matrix to the corresponding row of the target matrix is 0.5.
Therefore, the correct answer is:
D. [tex]\(0.5\)[/tex]
