Trivial Solution
A homogenous system of equations always has a trivial solution.
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Description
A homogenous system of equations has a vector of zeros on the RHS, as in:
┌ ┐ ┌ ┐ ┌ ┐ │ 1 2│ │ x│ │ 0│ │ 3 4│ │ y│ = │ 0│ └ ┘ └ ┘ └ ┘
Such a system always has a particular solution in the form of x=0 and y=0. This is called the trivial solution.
If a system is known to have only one unique solution and it is homogenous, it follows that the trivial solution is the one unique solution.