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For  n  = 1, the theorem is that, if  a  = 0, the equation  ax  = 0 has a solution for which  x  is not zero. This is obviously true. And the case of  n  variables can easily be made to depend on that of  n  – 1 variables. For brevity we show the method by taking  n  = 3.

Proof of the sufficiency of the determinant condition for the consistency of a system of n homogeneous linear equations In n variables