alleryFor any square matrix $A$, the matrix $A + A'$ is:
AAlways symmetric
BAlways skew-symmetric
CAlways singular
DAlways the zero matrix
Answer & Solution
Correct answer: A. Always symmetric
**Taking the transpose:** $(A + A')' = A' + (A')' = A' + A = A + A'$.
Since the matrix equals its own transpose it is symmetric.
By contrast, $A - A'$ is always skew-symmetric.
Every square matrix can be uniquely written as a sum $\frac{1}{2}(A + A') + \frac{1}{2}(A - A')$ of a symmetric and a skew-symmetric matrix.
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