= Solution
Let $M$ denote the displayed matrix of excess-return coefficients. If it were invertible, choose $v$ with $Mv=(1,0,0)^t$. Hold $v_i$ units of the three risky investments and short $\sum_iv_i$ units of the <risk-free asset>. Initial net cost is zero. The terminal payoff is
$$
\sum_iv_i(r_i-R)=\sum_iv_i(a_i-R)+\left(\sum_iv_ib_i\right)f_1+
\left(\sum_iv_ic_i\right)f_2=1.
$$
This is a guaranteed <arbitrage>, contradicting the assumption. Hence
$$
\boxed{\det M=0.}
$$
The contradiction does not require guessing probabilities for the factors: invertibility would allow simultaneous cancellation of both random exposures while keeping a nonzero constant gain.
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