= Solution
{c}
For $n\geq2$, take the matrix whose leading $2\times2$ block is $\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$ and whose other entries are zero. Its square has leading block $-I_2$, so
$$
\boxed{\varphi(A,A)=\operatorname{tr}(A^2)=-2<0.}
$$
This contradicts the <positive definiteness> required of an <inner product>. Even the nonzero matrix unit $E_{12}$ has $\varphi(E_{12},E_{12})=0$.
The printed assertion has an exception: \b[when $n=1$, $\varphi(a,b)=ab$ is an inner product] on $\mathbb R$. Thus the requested failure requires $n\geq2$, a restriction absent from the printed question.
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