= Determinant pairing is not an inner product
{title2=$B(A,C)=\det(AC)$}
On real <symmetric matrices> of size $n\ge2$, the displayed pairing is symmetric because $\det(AC)=\det A\det C$. It is not <bilinear>: $B(I,2I)=2^n$, whereas linearity would require $2B(I,I)=2$. Its self-pairing is always nonnegative, but a nonzero <singular matrix> has $B(A,A)=0$, so <positive definiteness> fails as well. For $n=1$ it reduces to the ordinary real <inner product>. The <Frobenius inner product> $\operatorname{tr}(A^TC)$ supplies the appropriate bilinear, positive-definite geometry in every size.
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