To solve this problem, we need to determine the value of the vector [tex]\(B\)[/tex] given the matrix [tex]\(A\)[/tex] and the resultant vector [tex]\(AB\)[/tex].
The system of equations given by [tex]\(AB\)[/tex] can be written as:
[tex]\[
A \cdot B = \left[\begin{array}{ccc}2 & 4 & -2 \\ 4 & -5 & 7 \\ 2 & 7 & 5\end{array}\right] \cdot B = \left[\begin{array}{r}24 \\ -46 \\ -2\end{array}\right]
\][/tex]
We need to solve for [tex]\(B\)[/tex].
Given the choices:
A. [tex]\(B=\left[\begin{array}{c}2 \\ 3 \\ -7\end{array}\right]\)[/tex]
B. [tex]\(B=\left[\begin{array}{c}1 \\ 2 \\ -7\end{array}\right]\)[/tex]
C. [tex]\(B=\left[\begin{array}{r}1 \\ 3 \\ -5\end{array}\right]\)[/tex]
D. [tex]\(B=\left[\begin{array}{r}2 \\ 2 \\ -5\end{array}\right]\)[/tex]
To determine the correct [tex]\(B\)[/tex], we can use a computational tool or numerical method to find the solution. After calculations, we find that the correct value for [tex]\(B\)[/tex] is:
[tex]\[
\left[\begin{array}{r}1 \\ 3 \\ -5\end{array}\right]
\][/tex]
Thus, the correct answer is:
C. [tex]\(B=\left[\begin{array}{r}1 \\ 3 \\ -5\end{array}\right]\)[/tex]
