Answer:
Statement c:
a₁₁ + ( b₁₁+ c₁₁ ) = 0
Step-by-step explanation:
The options are:
a: a₁₁ = 0 and b₁₁ = 0
b: a₁₁ - ( b₁₁ + c₁₁ ) = 0
c. a₁₁ + ( b₁₁+ c₁₁ ) = 0
d. a₁₁x( b₁₁ + c₁₁) = 0
We know that A, B, and C are matrices of the same size, so we can write:
[tex]A = \left[\begin{array}{ccc}a_{11}&a_{21}&...\\a_{21}&a_{22}&...\\...&...&...\end{array}\right][/tex]
[tex]B = \left[\begin{array}{ccc}b_{11}&b_{21}&...\\b_{21}&b_{22}&...\\...&...&...\end{array}\right][/tex]
[tex]C = \left[\begin{array}{ccc}c_{11}&c_{21}&...\\c_{21}&c_{22}&...\\...&...&...\end{array}\right][/tex]
And here we have the sum of matrices, remember that:
[tex]A + B = \left[\begin{array}{ccc}a_{11}&a_{21}&...\\a_{21}&a_{22}&...\\...&...&...\end{array}\right] + \left[\begin{array}{ccc}b_{11}&b_{21}&...\\b_{21}&b_{22}&...\\...&...&...\end{array}\right] = \left[\begin{array}{ccc}a_{11} + b_{11}&a_{21} + b_{12}&...\\a_{21} + b_{21}&a_{22} + a_{22}&...\\...&...&...\end{array}\right][/tex]
And we know that (A + B) + C = 0 (a matrix full of zeros)
then:
[tex]\left[\begin{array}{ccc}(a_{11} + b_{11}) + c_{11} &(a_{21} + b_{12}) + c_{12}&...\\(a_{21} + b_{21}) + c_{21}&(a_{22} + b_{22}) + c_{22}&...\\...&...&...\end{array}\right] = \left[\begin{array}{ccc}0&0&...\\0&...&...\\...&...&...\end{array}\right][/tex]
Then:
(a₁₁ + b₁₁ )+ c₁₁ = 0
These are real numbers, so we can rewrite this as:
a₁₁ + ( b₁₁+ c₁₁ ) = 0
Then statement c is the correct one.
Answer:
C
Step-by-step explanation:
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