Discussion
Let A and B be real symmetric matrices of same size. Which one of the following options is correct?
Answer: C
The correct statement is \((AB)^{T} = B^{T}A^{T}\).
The transpose of a product reverses the order of the factors; this holds for any matrices whose product is defined, symmetric or not. For symmetric \(A\) and \(B\) it also equals \(BA\).
The others need extra conditions. \(A^{T} = A^{-1}\) defines an orthogonal matrix, \(A = A^{-1}\) an involution, and \(AB = BA\) holds only for commuting matrices; for example \(A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\) and \(B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) are symmetric but \(AB \ne BA\).
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