Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 2 / 3 / a / Solution

A homogeneous linear system has a nonzero solution exactly when its coefficient matrix is singular. Expansion of the determinant givesThus the allowed values are $\boxed{\alpha =1\text{ or }-8/5}$. Solving the first two equations and checking the third givesIn each case $z\ne 0$ gives a nonzero solution and spans the one-dimensional null space; $z=0$ gives only the zero vector.