If [tex]$A, B$[/tex], and [tex]$C$[/tex] represent three matrices of the same size and [tex]$(A+B)+C=0$[/tex], then which statement is true?

A. [tex]$a_{11}=0$[/tex] and [tex]$b_{11}=0$[/tex]

B. [tex]$a_{11}-\left(b_{11}+c_{11}\right)=0$[/tex]

C. [tex]$a_{11}+\left(b_{11}+c_{11}\right)=0$[/tex]

D. [tex]$a_{11} \times\left(b_{11}+c_{11}\right)=0$[/tex]



Answer :

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]