 Solution

ID: 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 = 3 a - a^{2}

, or

a^{2} - 3 a + 2 = (a - 1) (a - 2) = 0.

This is also sufficient: the first two rows are linearly independent, and on setting the free variable

x_{3} = t

they give

a \in {1, 2}, (x_{1}, x_{2}, x_{3}) = (a - t, a^{2} - 2 a - 3 t, t), t \in R .

Substitution into the third equation gives

3 a - a^{2} = 2

for either allowed

a

. Explicitly the solution families are

(1 - t, - 1 - 3 t, t)

and

(2 - t, - 3 t, t)

. For every other

a

, the same row relation gives an immediate inconsistency.

 New to topics? Read the docs here!