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A matrix $M$ is symmetric if $M' = M$, and skew-symmetric if $M' = -M$.
We are given that $A + A'$ is a symmetric matrix. This means $(A + A')' = A + A'$.
Let's consider the transpose of $(A - A')$: $$(A - A')' = A' - (A')' = A' - A = -(A - A')$$ This shows that $(A - A')$ is a skew-symmetric matrix.
(A) (A - A') cannot be a skew symmetric matrix - This is incorrect, as we showed (A - A') is skew-symmetric. (B) (A - A') is a skew symmetric matrix - This is correct. (C) A is always a symmetric matrix - This is not necessarily true. (D) A is always a skew symmetric matrix - This is not necessarily true.
Final Answer: Option B<\/span>