Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 32 / 1 / c / ii / 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 ${\sum }_i{v}_i$ units of the risk-free asset. Initial net cost is zero. The terminal payoff isThis is a guaranteed arbitrage, contradicting the assumption. HenceThe contradiction does not require guessing probabilities for the factors: invertibility would allow simultaneous cancellation of both random exposures while keeping a nonzero constant gain.