Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Digital systems, linear algebra, introduction to formal proof / 4 / a / Solution

The third left-hand side is three times the first minus the second. Consistency of this linear system therefore requires $2=3a-a^2$, orThis is also sufficient: the first two rows are linearly independent, and on setting the free variable $x_3=t$ they giveSubstitution into the third equation gives $3a-a^2=2$ for either allowed $a$. Explicitly the solution families are $(1-t,-1-3t,t)$ and $(2-t,-3t,t)$. For every other $a$, the same row relation gives an immediate inconsistency.