To find the value of matrix [tex]\( B \)[/tex], we need to solve the system of linear equations given by [tex]\( A B = \left[\begin{array}{r}24 \\ -46 \\ -2\end{array}\right] \)[/tex].
Given the matrix [tex]\(A\)[/tex] and the result of matrix multiplication [tex]\(AB\)[/tex]:
[tex]\[
A = \left[\begin{array}{rrr}
2 & 4 & -2 \\
4 & -5 & 7 \\
2 & 7 & 5
\end{array}\right]
\][/tex]
and
[tex]\[
A B = \left[\begin{array}{r}
24 \\
-46 \\
-2
\end{array}\right]
\][/tex]
The system of linear equations represented by [tex]\(A B = \left[\begin{array}{r}24 \\ -46 \\ -2\end{array}\right]\)[/tex] corresponds to finding [tex]\( B \)[/tex] such that:
[tex]\[
\left[\begin{array}{rrr}
2 & 4 & -2 \\
4 & -5 & 7 \\
2 & 7 & 5
\end{array}\right]
\left[\begin{array}{r}
b_1 \\
b_2 \\
b_3
\end{array}\right]
=
\left[\begin{array}{r}
24 \\
-46 \\
-2
\end{array}\right]
\][/tex]
By solving these linear equations, we obtain:
[tex]\[ B = \left[\begin{array}{r}1 \\ 3 \\ -5\end{array}\right] \][/tex]
Therefore, the correct answer is:
C. [tex]\( B = \left[\begin{array}{r}1 \\ 3 \\ -5\end{array}\right] \)[/tex]
