To find the sum [tex]\( A + B \)[/tex] of two matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we add their corresponding elements. Given the matrices:
[tex]\[ A = \begin{pmatrix}
2 & -1 \\
5 & -2 \\
-3 & 4
\end{pmatrix} \][/tex]
[tex]\[ B = \begin{pmatrix}
6 & -1 \\
-6 & 5 \\
-1 & 0
\end{pmatrix} \][/tex]
Let's add them element-wise.
1. For the element in the first row, first column:
[tex]\[ 2 + 6 = 8 \][/tex]
2. For the element in the first row, second column:
[tex]\[ -1 + (-1) = -2 \][/tex]
3. For the element in the second row, first column:
[tex]\[ 5 + (-6) = -1 \][/tex]
4. For the element in the second row, second column:
[tex]\[ -2 + 5 = 3 \][/tex]
5. For the element in the third row, first column:
[tex]\[ -3 + (-1) = -4 \][/tex]
6. For the element in the third row, second column:
[tex]\[ 4 + 0 = 4 \][/tex]
Putting all these together, we get the resulting matrix:
[tex]\[ A + B = \begin{pmatrix}
8 & -2 \\
-1 & 3 \\
-4 & 4
\end{pmatrix} \][/tex]
Comparing this with the given options:
- [tex]\(\begin{pmatrix}-4 & -2 \\ -1 & -7 \\ -4 & 4\end{pmatrix}\)[/tex]
- [tex]\(\begin{pmatrix}-4 & 0 \\ 11 & -7 \\ -2 & 4\end{pmatrix}\)[/tex]
- [tex]\(\begin{pmatrix}8 & -2 \\ -1 & 3 \\ -4 & 4\end{pmatrix}\)[/tex]
- [tex]\(\begin{pmatrix}8 & 0 \\ -1 & 3 \\ -4 & 0\end{pmatrix}\)[/tex]
The correct option is:
[tex]\[ \boxed{\begin{pmatrix}8 & -2 \\ -1 & 3 \\ -4 & 4\end{pmatrix}} \][/tex]
