Solution to Exercise 3.17.
The good Lord made us with two ends–one to sit on and one to think with. How well you succeed in life depends on which one you use.
—Isaac Dworetsky

Exercise 3.17. The steps in the converse of the proof of Proposition 3.1 are reversible, but require that the nontrivial solution, [a₁, a₂, a₃], of the matrix equation cannot have both a₁ and a₂ be zero. Prove that this is true, which completes the proof of the proposition. 

Proof. Let [a₁, a₂, a₃] be a nontrivial solution of the matrix equation

 

We show that a₁ and a₂ cannot both be zero. Suppose they are both zero. Then

 

Hence, [a₃, a₃, a₃] = [0, 0, 0], i.e. a₃ = 0. Thus, [a₁, a₂, a₃] is a trivial solution, which contradicts that [a₁, a₂, a₃] is a nontrivial solution. Therefore, [a₁, a₂, a₃] is a line since that a₁ and a₂ cannot both be zero.//