I've thought about this question for about an hour but I'm still not able to arrive at correct answer. Can anyone suggest me how to go about it?
2026-04-19 12:58:39.1776603519
Similarity of real symmetric matrices
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We have, by definition of the adjoint $\def\<#1>{\left<#1\right>}$ $$ y^tBCx = \<S(x), y> = \<x,T(y)> = \<x,Ay> = (Ay)^tBx = y^tA^tBx$$ As this holds for all $x, y \in \mathbf R^n$, we must have $BC = A^tB$.