Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 18 / a / Solution

The matrix multiplication $A\mathbf{c}$ adds each row. Equating it to $\mathbf{b}$ gives the linear systemThe last equation is the sum of the first two, so it supplies no further restriction. Subtracting the first from the second gives $2\beta =2$, hence $\beta =1$, and then $\gamma =1-\alpha$. Thus all parameter choices are