 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

For real scalars $\alpha,\beta$ and matrices $A,C,B$, distributivity of <matrix multiplication> and linearity of the <trace> give
$$
\varphi(\alpha A+\beta C,B)
=\operatorname{tr}(\alpha AB+\beta CB)
=\alpha\varphi(A,B)+\beta\varphi(C,B).
$$
Thus \b[$\varphi$ is linear in its first argument].

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