Let's analyze the given equation step-by-step:
We have three matrices [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] of the same size, and the equation to solve is:
[tex]\[
(A + B) + C = 0
\][/tex]
1. Associativity of Matrix Addition:
Matrix addition is associative, which means:
[tex]\[
(A + B) + C = A + (B + C)
\][/tex]
2. Breaking Down the Zero Matrix:
The zero on the right side of the equation represents the zero matrix, where every element is zero.
3. Element-wise Form of the Equation:
Considering each element of the matrices [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex]:
[tex]\[
(A + B + C)_{ij} = 0_{ij}
\][/tex]
This means that for each element:
[tex]\[
a_{ij} + b_{ij} + c_{ij} = 0
\][/tex]
Specifically, focusing on the [tex]\( (1,1) \)[/tex] element of each matrix (i.e., the element in the first row and first column), we get:
[tex]\[
a_{11} + b_{11} + c_{11} = 0
\][/tex]
4. Rewriting in a Familiar Form:
We need to isolate [tex]\( a_{11} \)[/tex]:
[tex]\[
a_{11} + (b_{11} + c_{11}) = 0
\][/tex]
Therefore, the correct statement is:
[tex]\[
a_{11} + (b_{11} + c_{11}) = 0
\][/tex]
Hence, the correct option is:
[tex]\[ \boxed{a_{11} + (b_{11} + c_{11}) = 0} \][/tex]
