To find the value of [tex]\(c_{41}\)[/tex] in the matrix [tex]\(C = A + B\)[/tex], we need to follow these steps:
1. Construct Matrix [tex]\(A\)[/tex] and Matrix [tex]\(B\)[/tex]:
- Given:
[tex]\[
A = 2 \left[\begin{array}{ccc}
-6 & 7 & -4 \\
4 & -4 & 3 \\
-2 & 4 & 0 \\
7 & 0 & -2
\end{array}\right]
\][/tex]
[tex]\[
B = -3 \left[\begin{array}{ccc}
5 & -9 & 1 \\
3 & 6 & -3 \\
8 & 9 & 1 \\
4 & 9 & -9
\end{array}\right]
\][/tex]
2. Calculate Matrix [tex]\(A\)[/tex]:
- Multiply each element in the matrix by 2:
[tex]\[
A = \left[\begin{array}{ccc}
2 \cdot (-6) & 2 \cdot 7 & 2 \cdot (-4) \\
2 \cdot 4 & 2 \cdot (-4) & 2 \cdot 3 \\
2 \cdot (-2) & 2 \cdot 4 & 2 \cdot 0 \\
2 \cdot 7 & 2 \cdot 0 & 2 \cdot (-2)
\end{array}\right] = \left[\begin{array}{ccc}
-12 & 14 & -8 \\
8 & -8 & 6 \\
-4 & 8 & 0 \\
14 & 0 & -4
\end{array}\right]
\][/tex]
3. Calculate Matrix [tex]\(B\)[/tex]:
- Multiply each element in the matrix by -3:
[tex]\[
B = \left[\begin{array}{ccc}
-3 \cdot 5 & -3 \cdot (-9) & -3 \cdot 1 \\
-3 \cdot 3 & -3 \cdot 6 & -3 \cdot (-3) \\
-3 \cdot 8 & -3 \cdot 9 & -3 \cdot 1 \\
-3 \cdot 4 & -3 \cdot 9 & -3 \cdot (-9)
\end{array}\right] = \left[\begin{array}{ccc}
-15 & 27 & -3 \\
-9 & -18 & 9 \\
-24 & -27 & -3 \\
-12 & -27 & 27
\end{array}\right]
\][/tex]
4. Compute Matrix [tex]\(C = A + B\)[/tex]:
- Add the corresponding elements of matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
C = \left[\begin{array}{ccc}
-12 & 14 & -8 \\
8 & -8 & 6 \\
-4 & 8 & 0 \\
14 & 0 & -4
\end{array}\right] + \left[\begin{array}{ccc}
-15 & 27 & -3 \\
-9 & -18 & 9 \\
-24 & -27 & -3 \\
-12 & -27 & 27
\end{array}\right]
\][/tex]
This results in:
[tex]\[
C = \left[\begin{array}{ccc}
-12 + (-15) & 14 + 27 & -8 + (-3) \\
8 + (-9) & -8 + (-18) & 6 + 9 \\
-4 + (-24) & 8 + (-27) & 0 + (-3) \\
14 + (-12) & 0 + (-27) & -4 + 27
\end{array}\right]
\][/tex]
[tex]\[
C = \left[\begin{array}{ccc}
-27 & 41 & -11 \\
-1 & -26 & 15 \\
-28 & -19 & -3 \\
2 & -27 & 23
\end{array}\right]
\][/tex]
5. Extract the Value of [tex]\(c_{41}\)[/tex]:
- The entry [tex]\(c_{41}\)[/tex] corresponds to the element in the 4th row and 1st column of matrix [tex]\(C\)[/tex], which is [tex]\(2\)[/tex].
Therefore, the correct answer is [tex]\( \boxed{2} \)[/tex].
