Discussion
For any real symmetric matrix \(A\), the transpose of \(A\) is ____ .
Answer: D
The transpose of a symmetric matrix is \(A\) itself.
A matrix is symmetric precisely when \(a_{ij} = a_{ji}\) for all i and j, which is the statement \(A^{T} = A\). Transposing swaps rows and columns, and for a symmetric matrix that swap changes nothing.
\(A^{T} = -A\) defines a skew-symmetric matrix, not a symmetric one. \(A^{T} = A^{-1}\) defines an orthogonal matrix, and a symmetric matrix need not be invertible at all. The transpose of A is the null matrix only when A itself is the null matrix.
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